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SPELLING ABILITY 

ITS MEASUREMENT AND DISTRIBUTION 



B. R. BUCKINGHAM 



Submitted in partial fulfillment of the require- 
ments for the Degree of Doctor of Philosophy 
the Faculty of Philosophy, Columbia University 



PUBLISHED BY 

(Ztettlfmi OlaiUge, ©alumina 3lttterattn 

NEW YORK CITY 

1913 



SPELLING ABILITY 

ITS MEASUREMENT AND DISTRIBUTION 



B. R. BUCKINGHAM 



Submitted in partial fulfillment of the require- 
ments for the Degree of Doctor of Philosophy 
the Faculty of Philosophy, Columbia University 



PUBLISHED BY 

2tearl?er0 CfrilUg*, (Mumtria Intorattu 

NEW YORK CITY 

1913 



mm 



' 



k 



>w 



v 



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Copyright, 1913, by B. R. Buckingham 

Gift 

The University 

m I* 1814 



CONTENTS 

SEC. PAGE 

i Introduction. Purpose. Problem. Previous Investigation . . i 

2 Limitations 5 

3 The Original-List 6 

4 The Selected List. 18-Word List 8 

5 The Preferred Lists 11 

6 Examination of the First Preferred List 16 

7 Examination of the Second Preferred List 22 

8 Conclusions Regarding the Preferred Lists 25 

9 Ratings of Individual Pupils 27 

10 Overlapping 31 

1 1 Location of Grade Medians 34 

12 Scaling the Words 40 

13 The Use of the Scale 51 

14 The Zero-Point of Spelling Ability 55 

15 Observations on the Distributions shown in Fig. 21 61 

16 Supplementary Testing at Schools VI and VII 65 

17 Arrangement of the Words of the Preferred List by Teach- 

ers' Judgments 69 

18 Rice Sentence Test. Easy 50-Word Test 75 

19 Derived Forms of Distribution 84 

20 Conclusions no 

Appendix 113 



111 



INDEX OF TABLES 

NO. PAGE 

I Sample of word-ratings from Original (270) Word List. 

Schools I and II 8 

II Sample of word-ratings (118 words) 30-86. Schools 

III, IV, and V 12 

III Number correct, per cent correct, and rank of each 

word, First Preferred List 14 

IV Number correct, per cent correct, and rank of each 

word, Second Preferred List 15 

V Table to show method of deriving r-values by "foot- 
rule " method 18 

VI r-values between grades of School I (3 methods) 19 

VII Coefficients of Correlation, grade with grade, and each 
grade with all grades for each school (II, III, IV 

and V). First Preferred List 20 

VIII Correlations of school with school and of each school 

with all schools for each grade. First Preferred List 21 
IX Coefficients of Correlation, grade with grade and each 
grade with all grades for each school. Second Pre- 
ferred List 23 

X Correlations of school with school and of each school 
with all schools for each grade. Second Preferred 

List 24 

XI Distribution of individual ratings of pupils in Schools 

II, III, IV, and V 27 

XII Distribution of individual ratings grouped to show 

modes * 28 

XIII Number and per cent of pupils in each grade whose . 

ability equaled or exceeded that of the median pupil 

in every other grade 32 

XIV Table of values of the Normal Probability Integral 

corresponding to values of P.E 35 

XV The per cent of pupils in each grade whose ability 
equaled or exceeded that of the median pupil in 
every other grade with corresponding P.E. values.. 36 
XVI Direct and derived values of median distances in terms 

of P.E 39 

XVII Per cents and P.E. equivalents, Preferred List — all 

grades 45 

XVIII Grade positions and average positions 48 

XIX Words arranged in order of difficulty according to 
scale, their P.E. values and weights on a per cent 

basis 51 

XX A ten-point scale 52 

XXI Distribution of individual ratings. Easy 50-Word Test 57 

V 



VI 



Index of Tables 



NO. PAGE 

XXII Amount and per cent of overlapping with P.E. equiva- 
lents. Easy 50-Word Test 58 

XXIII Values of median intervals and their derivation (2 a to 

4th grade) 59 

XXIV Median Intervals o-8th grade 61 

XXV Distribution of individual ratings, Schools VI and VII. 

Selected List 66 

XXVI Comparison of results obtained in Schools VI and VII 

with those in Schools II, III, IV, and V 66 

XXVII Number and per cent of pupils in each grade who 
equalled or exceeded the median of every other grade 
with P.E.'s. Schools VI and VII combined with II, 

III, IV, and V 68 

XXVIII Median Distances derived from Table XXVII 69 

XXIX Comparison of results by Record and by Teachers' 

Judgments. Preferred List. . . .-. 72 

XXX Distribution of individual ratings. R. S. T 76 

XXXI Per cent correct for each word in each grade with P.E. 

values. R. S. T 78 

XXXII Per cent correct for each word in each grade with P.E. 

values. Easy 50-Word Test 80 

XXXIII Percentages of Retention — Grades 3-8 87 

XXXIV Plan of elimination and retention for each grade 89 

XXXV Derivation of 6th-grade Modified Table of Frequency . . 92 

XXXVI Modified Table of Frequency, 3d grade 93 

XXXVII Modified Table of Frequency, 4th grade 94 

XXXVIII Modified Table of Frequency, 5th grade 95 

XXXIX Modified Table of Frequency, 6th grade 96 

XL Modified Table of Frequency, 7th grade 97 

XLI Modified Table of Frequency, 8th grade 98 

XLII Number and per cent of pupils in each grade whose 
ability equalled or exceeded that of the median pupil 
in every other grade with the P.E. values corres- 
ponding to each per cent. Modified Distributions ... 10 1 
XLIII Direct and derived values of Median Distances. Modi- 
fied Distributions 102 

XLIV Comparison of Average Median Distances by Normal 

and Modified Distributions 103 

XLV Per cent correct for each word of Preferred List with 
corresponding P.E. values by Normal Distribution 

and by Modified Distributions 104 

XLVI Average position of each word by Normal Distribution 
and by Modified Distributions. Point of reference, 

3d-grade median 108 

XLVII P.E. values corresponding to given per cents of the 

Normal Surface of Frequency 116 



INDEX OF FIGURES 

NO. PAGE 

i Distribution of Individual Ratings. Selected List, 3d Grade 29 

2 Distribution of Individual Ratings. Selected List, 4th Grade 29 

3 Distribution of Individual Ratings. Selected List, 5th Grade 29 

4 Distribution of Individual Ratings. Selected List, 6th Grade 29 

5 Distribution of Individual Ratings. Selected List, 7th Grade 29 

6 Distribution of Individual Ratings. Selected List, 8th Grade 30 

7 Distribution of Individual Ratings. Selected List, All Grades 30 

8 Distribution of Individual Ratings. Rice Sentence Test, 6th 

Grade 33 

9 Distribution of Individual Ratings. Rice Sentence Test, 7th 

Grade. . , 33 

10 Distribution of Individual Ratings. Rice Sentence Test, 8th 

Grade t,^ 

11 Showing the Overlapping of the 3d and 4th Grade Surfaces 

of Frequency 34 

12 Normal Frequency Surface to Illustrate Word Placing 41 

13 Showing the Placing of the first seven words of the Pre- 

ferred List, 3d Grade 43 

14 3d Grade Scale. Preferred List 44 

15 4th Grade Scale. Preferred List 44 

16 5th Grade Scale. Preferred List 44 

17 6th Grade Scale. Preferred List 44 

18 7th Grade Scale. Preferred List 44 

19 8th Grade Scale. Preferred List 44 

20 General Scale, Preferred List 49 

21 Series of Grade Distributions (normal) showing Median In- 

tervals and the Zero-point. Entire range of Spelling 
Ability in the Elementary School 62 

22 Diagram showing difference in difficulty between words by 

Teachers' Judgments 73 

23 Distribution of Individual Ratings. Rice Sentence Test, 4th 

Grade 77 

24 Distribution of Individual Ratings. Rice Sentence Test, 5th 

Grade 77 

2 5 Scales for the following grades and lists : 

4th; Preferred, Rice Sentence, Easy 50-Word, 

5th; Preferred, Rice Sentence, 

6th; Preferred, Rice Sentence, n -p 

7th; Preferred, Rice Sentence, 1 i n g 

8th; Preferred, Rice Sentence J p. 80 

vii 



viii Index of Figures 

NO. PAGE 

26 2a Grade Scale, Easy 50-Word; 2b Grade Scale, Easy 50- ) 

Word j Fac _ 

27 3d Grade Scale; Preferred and Easy 50- Word I j ncr 

28 General Scale; (2a to 4th Grade combined) Easy 50-Word p. 82 

29 General Scale; Preferred, Rice Sentence Test, Easy 50-Word J 

30 Curves of Retention, 3d to 8th Grades 88 

31-36 The amount and distribution of elimination and retention. 

Grades 3 to 8 90 

37-42 Derived Forms of Distribution. Grades 3 to 8 99 

43-47 Comparison of Grade scales according to (a) Normal Dis- 
tribution and (b) Modified Distributions 106-107 

48 Comparison of General scales according to (a) Normal Dis- 
tribution and (b) Modified Distributions 109 



SPELLING ABILITY— ITS MEASUREMENT AND 
DISTRIBUTION 

§ i. Introduction 

The purpose of this dissertation is to derive a scale for the 
measurement of spelling ability and to show some of its uses 
and applications. Such a purpose relates itself closely to a 
general movement, which is now well under way, and which 
aims to place in our hands the means of stating with some- 
thing approaching the precision of objective measurement the 
amounts of each school ability possessed by an individual or 
a group. We received not long ago a scale for Handwriting 
(Thorndike, E. L., 1910) and still more recently a scale for 
English Composition (Hillegas, Milo B., 1912). The former 
consists in the use of selected specimens of handwriting each 
of which has been evaluated; the latter consists of a similar 
series of English compositions. It will be noticed that some 
of the conditions of objective measurement are met. We meas- 
ure given specimens of handwriting by comparing them with 
actual samples of handwriting of known value. We determine 
the quality of English composition by a like comparison with 
samples of actual English writing of known value. 

It seems clear, therefore, that if we are to measure ability 
in spelling at all it will be by reference to an evaluated standard 
or sample of spelling. If we can arrange a series of words on 
a linear projection in such a way that the words from the low 
end to the high end are placed at equal intervals determined 
by the difficulty of each word, and if we can determine a 
zero-point such that failure to spell the word fixed at that point 
under the required conditions indicates absence of spelling abil- 
ity, then we shall have constructed a scale by which we may 
measure the spelling ability of an individual, or by which we 
may through suitable tests determine the difficulty of any 
word in the language. Since the spelling of individuals may 



2 Spelling Ability — Its Measurement and Distribution 

thus be rated, the spelling of classes, of schools, and of school 
systems may likewise be rated. 

It may be said that we have always rated pupils in spelling; 
and that schools and school systems have likewise been rated. 
Such is indeed the case. But there has always been a lack of 
precision in these ratings due to the inequality of the units em- 
ployed. Dr. Rice (Rice, '97), for example, in testing the pupils 
in 4th to 8th grades in twenty-one school systems used a list 
of words containing among others: disappoint, necessary, 
changeable, better, because, picture. The method of rating 
pupils was the usual one of deducting from 100 per cent the 
same per cent for each word. That is, all words were taken as 
equal measures of spelling ability. A moment's attention to the 
six words mentioned will lead us to suspect that this is not a true 
assumption; and an actual test of a group of 5th-year children 
with these words shows that our suspicion is correct. In such 
a test mistakes were made as follows : 

disappoint, $7 

necessary, 42 

changeable, 42 

better, 3 

because, 1 

picture, o (Thorndike, '04, p. 8) 

To give these words equal weight in any test is to make 
inaccurate most of the conclusions based upon it. A pupil who 
spells all or nearly all of the list is a much better speller than 
the figures show; for he has probably spelled not only all the 
easy words but also many of the hard ones. On the other hand, 
a pupil who misses most of the words is a much poorer speller 
than his rating indicates because he has probably failed to spell 
all the hard words as well as most of the easy ones. 

Nor is this list of Dr. Rice's at all unusual. Cornman used 
the same list (Cornman, '02). Both used a composition test 
where pupils were rated according to the per cent of their 
correctly spelled words among the total number of words in a 
written exercise. Cornman also used a test in which school 



Introduction 3 

children were required to write " as many words as they could " 
in 15 minutes. Of course in the composition test and in the 
15-minute test no two children wrote the same words. More- 
over, the words written by each child must have varied widely 
in difficulty. The result for the 15-minute test, according to 
Cornman's best table, is as follows: 



School Year 


Median Percentage 


Average Variation 


8th 


97.9 


.60 


7th 


96.2 


.50 


6th 


95.2 


.33 


5tha 


94.3 


.36 


5th6 


94.3 


.10 


4tha 


94.7 


.66 


4th6 


93.7 


.96 


Sda 


93.5 


.23 


3db 


93.0 


1.43 



One conclusion from this is that " pupils of the elementary 
school increase regularly from grade to grade in accuracy of 
spelling." This might almost be taken for granted. But in 
answer to the question, " How much does one grade surpass 
another ?" the figures afford no information. Obviously from 
all we know of the elementary school, the difference between 
eighth-grade ability and low third-grade ability in spelling is far 
greater than the figures 97.9 and 93 indicate. 

Similarly the Composition Tests of Rice and Cornman are 
misleading when used to indicate spelling ability. According -to 
the series of Composition Tests of the latter, 8th-year children 
on the average spelled 99.5 per cent of their words correctly, 
and children of the first half of the 3d year spelled 93.2 per cent 
correctly. The author draws conclusions from his figures as 
to the progress of each grade for the school year, as to the 
progress of the school and as to the effect of the suspension 
of instruction in spelling. Since in the series of eight tests 
the children wrote various kinds of lessons — Geography, History, 
Science, Language, Composition — each with its own peculiar 
words, and since each pupil used his own individual vocabulary, 
we cannot escape the conviction that while these figures may 
be suggestive of progress, or of the effect of change in method, 
or of grade differences, they are nothing more than suggestive. 



4 Spelling Ability — Its Measurement and Distribution 

They leave unanswered the questions, — How much progress? 
How large an effect? How great a difference? As we grow 
more and more accustomed to quantitative thinking in our edu- 
cational work, we feel that these are precisely the questions that 
we ought in some way to be able to answer. 

These studies of spelling made by Cornman and Rice remain 
the most important statistical treatment of the subject. That 
they have not great value it would be presumptuous even to 
imply. Their results are in a general sense true. To a certain 
extent their lists, even though made up of words of various- 
and undetermined difficulty, may be used, especially for com- 
parative purposes, as a total measuring device. They do, how- 
ever, undoubtedly suffer through lack of precision, while their 
statements of amounts of difference are in general misleading. 

The same thing may be said of later investigations. For 
example, Wallin's tables and his conclusions from them as to 
the transfer of spelling efficiency and its relation to age, grade, 
and sex are subject to the same limitations (Wallin, 'n). Pear- 
son's " Experimental Studies .in the Teaching of Spelling " 
(Pearson, '12), however, shows a recognition of the difficulty, 
although it offers no remedy. In his treatment of the relative 
values of the " together-method " and the " separate-method " 
of teaching homonyms this author says : " Owing to the in- 
equality of the units of measurements, it is impossible to deter- 
mine accurately from Table IV whether the together-method 
is superior to the separate-method. One cannot decide, for 
example, positively whether an improvement from 3.78 errors 
(median of a class) to 2.86 errors is greater or less than an 
improvement from 5.6 errors to 3.3 errors." If, however, the 
words used could have been evaluated through an independent 
test by reference to a scientifically constructed scale, the " in- 
equality of the units of measurement " would have disappeared. 
The further treatment of the foreshortening of the opportunity 
for improvement due to high initial performance is quite another 
matter. 

It will be clearly seen from the foregoing that in practically 
all work which has attempted to present the spelling situation 
statistically it is assumed as fundamental that one error equals 



Limitations 5 

another and that to spell one word is the same as to spell 
another word. 

It will therefore be profitable to seek in this field as others 
have sought in other fields to devise an instrument which will 
more accurately measure that of which we are so often called 
upon to give a quantitative statement. 

§ 2. Limitations 

The study here attempted is confined to the elementary school 
entirely. It covers the grades from the third to the eighth, both 
inclusive. The schools tested are all located in or near New 
York City. The cosmopolitan character of the population of 
the metropolitan area makes it extremely unlikely that results 
of a materially different character would have been obtained 
by testing schools in various sections of the country. 

It is believed that these schools are fairly typical within the 
limits of the area chosen. School I is a private school of high 
class whose pupils are mostly American born and from good 
homes. All the other schools are public schools. School II 
is in a German section of rather low class. School III is in a 
better neighborhood, foreigners predominating. School IV is 
in an Italian section. It has long had the benefit of high-class 
supervision and organization. School V is again predominantly 
American. It is located outside of the city system. School VI 
is in a good residential section of the city. School VII is a 
large school, most of whose pupils are of foreign parentage. 
Territorially, two schools are in Manhattan, one in the Bronx, 
one in Brooklyn, and two in Queens, while one is outside of the 
city entirely. 

In all 8,791 pupils were tested. It is thought that this is a 
sufficient number for practical purposes. In fact it was found 
that the returns from each additional school after the first three 
or four made almost no change in the results. It is probable 
that greatly increasing the number of pupils tested would have 
afforded little compensation for the additional labor. It has 
seemed wiser to limit the number to a moderate one and to 
spend considerable effort in making the statistical analysis as 
complete as possible. 



6 Spelling Ability — Its Measurement and Distribution 

§ 3. The Original List 

The preliminary testing was made with a list of 270 words. 
It will be called " The Original List." It was itself selected 
from a much larger list of graded words used by the author 
of this dissertation in his own school, the same having been 
secured by taking from five of the popular Spelling Books now 
in use a vocabulary of 5,000 words agreed upon by two or more 
of the books. The principles of selection for these 270 words 
were: (1) that all of them should be sufficiently common to be 
in the speaking vocabulary of third-grade children; and (2) that 
the spelling difficulty of many of them should be great enough 
to test the ability of eighth-grade children. As a matter of fact, 
the selection did not consist of 270 words at first. The list grew 
to that number only after the chosen words were put into sen- 
tences. The necessary helping words then swelled the total to 
the number given. 

The sentences were dictated during the fall term of 1910 to 
schools I and II. They were given to grades 3 to 8 in School 
II, and to grades 4 to 7 in School I. Their dictation consumed 
several periods for every class. The following are the sentences : 

There were forty birds on the bridge. Do not go until I come. 
On Wednesday an umbrella was found. Whose pencil is this? 
My uncle gave me a banana. The butcher gave the hungry dog 
a piece of meat. My answer is ninety. For a nickel I bought 
an orange, a peach, and a pear. A dollar is not too much money 
for so beautiful a picture. Learn to do right because it is right. 
The chicken ran across the road. The janitor sweeps every Tues- 
day afternoon. It is wrong to steal even a penny. It would be 
easy to watch for your cousin from the parlor window. It is 
the hour for recess. Smoke was coming out of their chimney. 
One summer evening my neighbor came into my kitchen. I did 
not know he was coming that night. To whom does this pair of 
scissors belong? I am almost sure they belong to the tailor. 
The doctor thought he ought to go at once. His bicycle was 
against the fence. But a carriage was stopping in front of his 
office. His friend was already beginning to speak to him. He 
said the soldier should have medicine this minute. Pshaw, there 
was neither a monkey nor an elephant at the circus. Get some 



The Original List 7 

coffee, sugar, and soap at the grocery store. The soldier dropped 
his sword and pistol. Jack had a whistle and nineteen nails in 
his pocket. The pretty fairy had a sawcy tongue. One rfa^; in 
February people saw a. sleigh pass through the avenue. Shoes 
are made of leather and a Zitffe ^Yow. A weefc from to-day there 
ze/i// be a dance. Cut up a tomato and an omow together. In my 
garden I ^Aa// raw^ cabbage instead of &££fo. The saucer was 
round like a circle. Make no noise; do not whisper or laugh. 
Nobody should be without a. handkerchief. A straight line has 
length only. We shall believe the frwfA. We have another piano 
at cwr school. Is it Jrwe that there was grease on the towel? 
This animal has a /arg^ mouth. It is not o/ten co/d enough for 
the ocean to freeze. Guess what made me sneeze. Choose which 
one of the pigeons you like. Touch the button with your thumb. 
The American Indian had com and tobacco. I have written the 
whole alphabet. I wear a number thirteen collar. If the m^w 
quarrel, telephone me or .send a telegram. Our arithmetic lesson 
is in addition. We a/so subtract. A handful of corn was a// I 
had for supper. What is the £i£/c of the story f Did you /^or 
the thunder last night? I am fywo; up my shoe. A &<mw of 
water sat on the fa&te. That sentence has twelve words in it. 

Those who dictated the sentences were directed to read them 
in whole or in part as many times as seemed necessary to secure 
their complete comprehension. Pupils were therefore not re- 
quired to retain in mind a long series of words. 

In rating the papers only the words printed in italics were 
considered. If a word occurred twice it was regarded only the 
first time it appeared. Omitted and illegible words were classed 
as wrong. All the papers here as well as elsewhere throughout 
this study were rated by the same person. They were rated 
from two points of view: (1) as to the number of times each 
word was correctly spelled, and (2) as to the per cent of the 
entire number of words each pupil spelled correctly. The former 
point of view is the only one to which attention is now directed. 

Table I is a sampling from the entire 270 words as given to 
schools I and II. At School I the grammar-school course is 
completed in seven years. It therefore has no 8th grade. As 
stated above, the test was not given to the 3d grade in this school. 



8 Spelling Ability — Its Measurement and Distribution 



TABLE I 

Figures Indicate Per Cent Correct 

Table reads: "across" was spelled correctly in the 3d grade of School II 
by 17% of the pupils; in the 4th grade of School I by 60% of the pupils, 
and of School II by 40% of the pupils, etc. 



Grade 

School . . . 

across 

addition. . . 
almost 
alphabet . . 
arithmetic. 

bridge 
button 
choose 

day 

guess 

handful . . . 
pshaw 
tomato... . 

too 

whose 



3d 


4th 


5th 


6th 


7th 


II 


I 


II 


I 


II 


I 


II 


I 


II 


17 


60 


40 


76 


58 


90 


79 


98 


87 


2 


38 


26 


60 


28 


76 


45 


94 


76 


16 


62 


41 


73 


65 


88 


75 


80 


81 


25 


13 


1 


63 


12 


40 


46 


82 


43 


27 


89 


53 


100 


72 


96 


92 


100 


97 


29 


59 


42 


87 


52 


98 


85 


100 


94 


14 


50 


35 


70 


49 


77 


63 


84 


62 


6 


25 


10 


37 


31 


62 


37 


67 


55 


97 


100 


98 


96 


100 


100 


99 


100 


100 


6 


29 


17 


67 


30 


77 


50 


82 


66 


36 


47 


33 


46 


19 


76 


33 


75 


63 


1 


4 


6 


29 


6 


46 


5 


31 


31 


34 


83 


49 


67 


43 


74 


48 


79 


32 





10 


3 


17 


4 


26 


7 


63 


22 


17 


49 


15 


40 


29 


47 


10 


57 


59 



8th 



II 



93 

83 
87 
68 
98 

97 

83 

65 

100 

85 

57 
18 
38 

27 
66 



§ 4. The Selected List 

On the basis of the results for the Original List, a group of 
100 words was chosen. It is here called the " Selected List/' 
In Table I are shown 15 words from the Original List. The 
word " across " is typical of the words taken for the Selected 
List. Since 17 per cent of the 3d-grade children spelled it cor- 
rectly, it was not so difficult in that grade as to offer no test 
of ability. It showed a steady increase throughout the following 
grades but did not reach so high a figure in the highest grades 
as to prevent its being a test of ability there. "Almost" and 
" button " were chosen for the same reason. "Addition " was 
not taken because it was too hard for 3d-graders. Only 2 per 
cent wrote it correctly. So small a number as two in a hundred 
might get it right by chance. Practically, therefore, the word 
is a zero word for the 3d grade; and such a word does not test 
ability. There may be — and in a given grade there certainly 
would be — wide differences in spelling ability, but such a word 



The Selected List 9 

will not show them. "Alphabet" was rejected because though 
high in the 3d grade it was very low in the 4th, suggesting that 
in School II it was a word that the children had recently studied. 
"Arithmetic " was not taken because from the 6th grade on it 
offered practically no difficulty. As in the case of a word rated 
at zero or nearly zero, so in the case of a word rated at 100 
or nearly 100, there is no test. Good spellers and poor spellers 
so far as the particular word is concerned behave exactly alike. 
" Bridge " was not taken for the same reason. " Choose " was 
too hard in the 3d grade. " Day " was too easy everywhere. 
In fact " day " is a type of word such that we may almost be 
warranted in saying that one who cannot spell it has no spelling 
ability. " Guess " was taken because although it is very seldom 
spelled correctly in the 3d grade, its form is so peculiar that the 
few who did write it correctly probably knew how to spell it, 
i.e., did not get it right by chance. " Handful " is a type of 
word taken because although it shows no regular increase from 
grade to grade it offers a real test for every grade. The later 
results in other schools, however, showed that its irregularity 
is not accidental in schools I and II but is a peculiarity of the 
word itself. " Pshaw " is a familiar word to the ear, but not 
to the eye. Very few get it right in any grade. It was rejected. 
" Tomato " is curious. On the whole neither school does any 
better with it in the highest than in the lowest grades. It was 
not taken. This word and the word " handful " strongly suggest 
the need of a greater number of pupils to test. The word " too " 
is a word which is misspelled with astonishing frequency. The 
difficulty is not so much one of spelling as of confusion with the 
other two words which have the same pronunciation. It was 
not used in the Selected List but was later included in a small 
supplementary list just to " try it out." " Whose " was taken 
although it shows a dip in the 5th and 6th grades. Pupils in 
these grades have learned the use of the apostrophe and their 
" little knowledge " proves a " dangerous thing " which the pupils 
of the earlier grades avoid. 

These words — each more or less typical in its way — show how 
from the Original List of 270 a better Selected List of 100 was 
chosen. Again the words were put into sentences, as follows : 



io Spelling Ability — Its Measurement and Distribution 

Whose answer is ninety? If the janitor sweeps, he will raise 
a dust. You ought not to steal even a penny. Wait until the 
hour for recess to touch the button. Smoke was coming out of 
£/^ir chimney. Every afternoon the butcher gave the hungry 
dog a />i£T£ of meat. One evening a carriage was stopping in 
/r<w£ of my kitchen. I w^ar a number thirteen collar. Guess 
what made me sneeze. Send me a />air of leather shoes. I do 
not know, but I am almost sure they are mine. My wncte bought 
my cousin a pretty watch for /arty dollars. The soldier dropped 
his sword. Jack had a whistle and a/so twelve nails. The oceaw 
do*?.? not o/fe« freeze. You should speak to people whom you 
meet. It takes ow/y a minute to />a^ through the gate and across 
the road. Did you ever A^ar a /airy laugh f The American 
Indian had a saucer without a cup. Neither a />^ar wor a peach 
was at the grocery store to-day. Cut up a w/wfe owiow with a 
handful of beans. My />iawo lesson was ea^sy. The animal ran 
wfo the road and straight against a tree. Give me another sen- 
tence which has the word " ftf/* " in it. I believe true friends 
like to be together instead of apart. 

These sentences were dictated at schools III, IV, and V in 
the spring term of 191 1. They were later (fall of 1912) dictated 
at schools VI and VII. 

The following instructions were given to the examiners : 

Please read these instructions through before beginning to dic- 
tate the sentences. 

I. See that each sheet is headed with (a) the pupil's name, 
(b) the date, (c) the grade, (d) the name of the school. 

II. Give all the sentences during one session, i.e., either in 
the morning or the afternoon of the same day. 

III. In classes below the fifth year dictate in two periods, 
separated by at least half an hour, or by a recess period. 

IV. Each sentence may be dictated, either in whole or in part, 
as many times as may seem necessary to secure its complete 
understanding. This exercise is purely a test in spelling. It is 
not intended that pupils should be subjected to the added diffi- 
culty of an effort to recall the words dictated. 

V. Offer no explanation of words or sentences. If the mean- 
ing is not clear, repeat the sentence as a whole or in part. 

VI. Do not ask the children to underline words nor otherwise 
call their attention to the significant words of the sentences. 



The Preferred Lists n 

VII. After the children have written the sentences, read them 
again and allow pupils to insert words or make other corrections. 

VIII. Collect the papers. 

Subsequently at the same schools (III, IV, and V) was given 
a supplementary list of 18 words, again selected from the 
Original List (270 words). With the same directions to the 
examiners, these words were put into sentences as follows : 

Telephone me on Tuesday if the tobacco comes. The tailor 
sent a saucy telegram. Already the circus was beginning. 
Pigeons seem too beautiful to quarrel. I am trying to choose a 
towel. The chicken was fried in grease. 

Each of these 118 words was scored in each grade and for 
each school separately. Table II illustrates for a few of the 
words the manner in which this was done. The figures indicate 
per cent correct. 3a means third grade, first half ; 3Z? third grade, 
second half, etc. Ill, IV, and V refer to different schools. 

It will be seen at once that there is no steady progression for 
each word as we pass from the lowest to the highest grades. In 
fact for this and for other reasons it was found best to deal 
with grades by years rather than by half-years. It also seemed 
advisable to choose a few of these words and to make them 
the basis of study. 

§ 5. The Preferred Lists 
From the data now in hand it was possible to select a few 
words which showed reasonably regular increase from grade to 
grade in the per cent of times they were spelled correctly. Two 
lists were made up, each containing twenty-five words. The 
first list is superior to the second in the testing power of the 
words in all grades and in the permanence of their relative diffi- 
culty throughout the grades. That is, to a somewhat greater 
extent than in the case of the second list, the words of the first 
list are found to be easy enough for low grades and hard enough 
for high grades. Also, a word occupying a certain serial posi- 
tion (say the 4th in point of difficulty for the third grade) tends 
more strongly in the first than in the second list to occupy the 
same position in all other grades. That both lists, however, are 
reasonably satisfactory in these particulars will be abundantly 
shown. 



_ — — - 



12 Spelling Ability — Its Measurement and Distribution 



TABLE II 

Figures Indicate Per Cent Correct 



Grade , 

against 

III.. 

IV.. 

V.. 

believe 

III.. 

IV.. 
V.. 
cousin 

III.. 

IV.. 

v.. 

know 

III.. 

IV.. 

V.. 

ninety 

III.. 

IV.. 

V.. 

pigeons 

III.. 

IV.. 

V.. 

saucer 

III.. 

IV.. 

V.. 

too 

III.. 

IV.. 

V.. 



3a 


36 


4a 


46 


5a 


56 


6a 


66 


7a 


76 


20 


60 


54 


43 


60 


62 


60 


91 


73 


93 


5 


19 


26 


29 


70 


61 


92 


78 


95 


94 








10 


15 


30 


52 


54 


74 


63 


81 


8 


77 


37 


54 


53 


44 


35 


57 


73 


93 





3 


17 


36 


32 


22 


50 


52 


62 


78 





4 


24 


14 


28 


55 


76 


50 


77 


71 


12 


16 


44 


72 


78 


83 


91 


95 


98 


97 


10 


39 


38 


58 


72 


82 


97 


93 


97 


96 


11 


28 


45 


38 


55 


68 


84 


74 


68 


90 


22 


40 


63 


75 


73 


70 


88 


82 


93 


97 


37 


61 


68 


71 


78 


81 


85 


93 


97 


98 


33 


38 


62 


60 


60 


84 


80 


79 


79 


88 


56 


61 


65 


46 


60 


65 


70 


86 


71 


80 


15 


25 


54 


19 


59 


49 


61 


60 


87 


82 





11 


28 


14 


36 


52 


64 


61 


62 


71 





2 


13 


31 


24 


61 


66 


49 


83 


64 


8 


6 


74 


50 • 


63 


22 


57 


67 


81 


74 





45 


9 


32 


61 


50 


82 


64 


50 


89 


7 


4 


17 


52 


45 


40 


33 


73 


78 


83 


10 


10 


16 


36 


49 


51 


73 


60 


83 


72 





16 


31 


19 


21 


52 


66 


45 


66 


79 








15 


58 


64 


46 


14 


60 


24 


14 


3 


45 


34 


27 


54 


40 


10 


11 


33 


33 





45 





85 


18 


26 


36 


41 


18 


26 



8a 86 



97 95 
100 97 



73 

83 
84 



97 
83 
79 

96 
84 
80 

80 

78 
81 

65 
53 
51 



85 
78 
72 



100 100 

100 100 

86 91 

97 100 

98 100 
100 96 



80 
86 
74 



95 

84 

75 

78 
78 

32 
45 
39 



The first list will be called the " First Preferred List." It 



contains the 


following words : 




i. even 


10. forty 


18. saucer 


2. lesson 


11. pretty 


19. stopping 


3- only 


12. wear 


20. sword 


4. smoke 


13. button 


21. freeze 


5. front 


14. minute 


22. touch 


6. sure 


15. cousin 


23. whistle 


7. pear 


16. nails 


24. carriage 


8. bought 


17. janitor 


25. nor 



9. another 



The Preferred Lists 



13 



The second list, called the " Second Preferred List," is as 
follows : 

10. tailor 

11. telegram 

12. telephone 

13. tobacco 

14. too 

15. towel 

16. Tuesday 

17. tying 



18. whole 

19. against 

20. answer 

21. butcher 

22. guess 

23. instead 

24. raise 

25. beautiful 



1. already 

2. beginning 

3. chicken 

4. choose 

5. circus 

6. grease 

7. pigeons 

8. quarrel 

9. saucy 

Table III gives for each word of the First Preferred List 
and for each grade the number of times the word was written, 
the number of times it was spelled correctly, and the per cent 
correct. Schools I, II, III, IV, and V are included. (Omitted 
words are considered as " written " and as wrong.) Table IV 
gives the same facts for the Second Preferred List. 
/It will be seen from tables III and IV that for any given 
word the per cent correct in one grade is higher than it is in 
any lower grade. This is, of course, to be expected. But it 
is not sufficient. In order that this list should be of greatest 
value it should be so constituted that these increases in 'per- 
cents-correct ' so keep pace with the increase from grade to 
grade of general spelling ability that a word tends in all grades' 
to maintain the same difficulty relative to all other words in the 
list to which it belongs. A word which is 20th in point of diffi- 
culty for the 3d grade ought to deviate as little as possible from 
the same rank in the other grades. The experience gained in 
making this investigation leads us to think that most words 
do not meet this condition even approximately. The span be- 
tween the 3d and the 8th grades is very wide. Accordingly a 
very large class of words is impossible for the earlier, yet easy 
for the later, grades. Still others are really difficult in the lower 
grades but of almost no difficulty in the upper grades. From 
our own Original List " coffee " and " people " are hard for 
3d- and 4th-graders, but are almost always spelled correctly 
above the 6th grade. A third group of words breaks down in 
the middle. They appear to be easy in the lowest and highest 



14 Spelling Ability — Its Measurement and Distribution 





H 




m 




H 




H-l 


►— 1 


ft 


HH 


w 


t— 4 


K 


m 




hJ 


h 


pq 


H 
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Ph 

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to 






d J4 MI'S 



3{UBy: 



k5 






3{UT3H 



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sz§8 






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h» O <D 



o\i$T5 






H-*<N <N*I> 

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t^CNiO ©Oh -*CO"* CO^HO"* 
000505 050500 00O5O5 05 X 00 OS 



C5CO-* CO00CN 
IOBC COiOlO 
NNN CN(N<M 



CO"3 HHOJ 

mom eoio-* 

(NNN (NCNCN 



(NIC* 
CO*iO 
CNCN<N 



C0 05 10 

ioeoi-1 

<M<N(N 



1«1>H t-~CDCDH 
<N*iO *(M<NiO 
(MCNCN (MCNcMcM 



b-l^t>- l>t>t> l>t^l> t^t-t> I>l>b- l>t>t> l>l>b- I>t^l>b- 

COCOO OCDCD ffitOffl CO CO CD CO CO CO CO CD CO CD CD CD CD CO CO CD 
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C0-*t> 



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lOSN 1O0000 l>COt^. CDb-CO 
050505 05 00 00 050505 05 C5 05 
COCOCO COCOCO COCOCO COCOCO 



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OOOOlO -tfOCOOOO 
000505 05050005 

COCOCO cooococo 



CO*H 



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CN*lO 

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NHM 
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05(N»0 



OHO 00CO00 
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COCO* COCOCM 



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COCOCO COCNCOCO 



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lOiOlO iO co>o 



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COOCO 



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05CM05 
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COCDlO C005t-H 
COCOCO COCOCO 
lOiOlO lOiOlO 



CD»0O 



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t^oot> 

COCOCO 



05 05 10 



lOOlO 
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CM CO CO CO CO <M CO 



00O5t- 1>C50000 
COCOCO CO CO CO CO 



■*iOLO 

oo-* CO 



kOOOlO 

00l>00 



lOeor-i 



cooot^ 
co-* co 

COCOCN 



Ot^OS 00COIO 



COOCO "HCOCD 
0<MO OHM 
CNCOM COCOCN 



co'-*co 



* lOlO 

cot^t^l 



lO»OI> COOSCO C5"*CO 
OOOOOC 000000 00CO00 



00iOI> COCM'-H i-i CON <M-*-*i-H 
CO00CNJ COO* OrHCO *IOC500 
IMi-lN NNH hNN (NcNtH'N 



i-H * 00 *lO>D ID*'* i-H*lOi-H 
CO0000 COX00 OOCOCO CDCOOpCD 



cotNi-i t»*co coHco iooo* 



CO*iO 
»OiOi> 

CN-HCN 



CNiMCM 



N>00 
OHO 
HCNCM 



C0*iO 
NNN 
■*** 



HlOH 
CM CM CM 



05005 
005* 



IDIO-* 
CMCNCN 



CDCOlO 
CNCNCN 



»OC0(M 
CMCNCM 



*iOiO 
CNCNKN 



OlO*H 

CNCNCNCM 

TjH Tf ^ Tf 



a . 
£ ! S >» 



S E S »oc) 

m«5 m Q~0 cS 



>.§ 



a^ JoS 



§|.S ro :ofe 

■S.5P ■ 



bS) 



"S.-0 d ja'53 i 
»oS 1-2'g^ 

03 0Q<S *>?OC 



The Preferred Lists 



15 









lO 


10115 


1O1O1O 


»o »o 


»OkO 


1010 m 


lOiOlO 




5[UBH 


NCOH 
NN 


00"C* 

H N 


OOOH 

rH N 


OO"* 


■tfiOOO 
N 


r-lHCO 


XNN 


Oi-hoo-* 




*S 


0»00 
OOO 


(MiON 
OOOiO 


N"*00 
QOOO 


Ti<<*o 

00 00 00 


coco-* 

05^05 


hOO 
OXC5 


■*oo 

000 


>OHr}tD 

XOOO 


w 

00 


6 >- O 
& u 


ONO 

0010 

H>-I<N 


OlO CO 

HTt<H 

NNH 


— 1H0 
NNN 


ooco 

Hr-KN 

NNN 


OOHH 
H,-|H 
N<-H<N 


CO^Ci 
COCNCO 
CNCNCN 


HOO 
lO lO lO 
CNCNCN 


00 CN CNOO 
Nhioh 
CN CNCNCN 


No. 
tak- 
ing 
part 


OOO 

iO iO iO 
NNN 


OOO 

»OiO>0 
NNN 


OOO 

lOiO^O 
NNN 


OOO 

1O1O1O 
NNN 


NNN 

LQiOiO 
NNN 


IO lO CD 
NNN 


OOO 

OOO 
CNCNCN 


(0(00115 
CN CNCNCN 
















lO O 


lO »C IO 


LO 




^m?^ 


COt-Ii-I 


NO 1 * 

NHlN 


OOO 
CM —1 


COOON 


COIO-* 
N— 1 


NOO 


O^Tt 1 


l-OOCON 


PJ 
< 


{£ 


•*oo 

■*0O 


lOiON 
cooh 


OOH 
OOOO 


HCOCO 
00O00 


00O00 
XCOO 


OO"? 
XOX 


*oo 

CC OO 


O00OO 





O >H O 

•^ O 0) 


■^i-iOO 

lO CO CO 
HNCO 


ONO 

(MOH 


rfOOO 
NOON 


OrflO 
OlOOO 
NNN 


IO TH— ( 
OON 
COHN 


CO CO CO 
OCOCO 
COCNCO 


CNOO 
COOlO 


CNOOO 

O-^OCN 
CO CO CO CO 


No. 
tak- 
ing 
part 


NNN 

1O1O1O 
CO CO CO 


OOr-100 

CO CO CO 


OOOH 
HiOLO 
CO CO CO 


NOO 

TfLOlO 


xo x 

CO CO CO 


OOO 
CO CO CO 


iO>OiO 
CO CO CO 


coooo 

OOO"* 
CO CO CO CO 










10 


»o to 


*c 




10 10 


115 »o 10 




3fUB^J 


OCOH 
rHN 


O^n 
NhN 


HON 
NHN 


O00 CO 


010 CO 

1-HNH 


OiOX 


CNH 


ONO"# 


« 


£? 


NOO 
Ot^o 


ON>0 
OOCO 


OlON 

1OO1O 


»o COO 

OOO 


IC*C0 
ONO 


00000 
00 coo 


IOOX5 
00c 00 


NCO^lfl 

oxxx 


H 

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10(00 
NON 
NHCO 


o-^oo 

HCON 
NNH 


OONH 
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NiMH 


CO oo-* 

ON"# 
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tJ((ON 
000 CO 

CN CN 


CNCO00 
OHIO 
CM IN CO 


000 

■*oooo 

CO CO CO 


OCOOh 
OOMh 
co co co co 


No. 
tak- 
ing 
part 


00(0 


lO COLO 
CO CO CO 
CO CO CO 


LONCO 

CO CO CO 


H,MN 
CO CO CO 
CO CO CO 


1010>0 

CO CO CO 
CO CO CO 


lO (NO 
CO COO 
COCOH 


00 CO CO 
lOiOLO 


OOOiO 
1010K5CD 
H H H CO 






O fc o 

> O J) 
<i O Z 









hr O 0) 

<5 o Z 

<<S— 1 o. 






o h o 






CNiOO OCOO COOiO OhN« 



CONN OOO OO-CH OQOO 
00OO O^CO ONi> COO 
HrtM NNrH HNtH COHN 



NOO 

Oh 00 
NH(N 



HlOO 
OOH 

NCOCO 



NCOCO NCCCOCO 



OOO 



0000 

CO CO CO 






OOO OOO OOO 00OO OiOiOO 
COCOCO COCOCO CO coco cococo cocococo 
t^hm^ •htJhiO lOiOiO lOiOiOH 



N-*0 lOOO 



lOCO 00000 



NHCO H 



OON Hr-i— 1 OOO 0<*i0 ONO OOOOO OHHO 
IO0000 COOO OHco OCOO rfXO f NCO HcoOiO 
HHr-l NtHtH HH(N CONN rHNN HNNN 



COOO 
00 GO 00 



COCOCO COTtHCO O-"*-* COOCO OOO OOO NrfiOCO 
OCOO OOO CO 000000 O00O OOOCiO 00X00 oxxo 



oxx 

Nh 
CN-*h 



HCOH 

N N N 



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NNNO 

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S§= S|-38 

rt rtX2 bc.5 "-2 



1 6 Spelling Ability — Its Measurement and Distribution 

and hardest in the middle grades. " Whose " is a type of such 
a word. It presents no great difficulty until children learn the 
use of the apostrophe. Then they write " who's." Later they 
partly recover from this practice. There are quantities of words 
which show this dip in the middle grades. One homonym is 
often easy until the other has been consciously related to it. 
Analogies falsely assumed play a harmful role. The rapid en- 
riching of the vocabulary as new subjects and new phases of 
old subjects are taken up in the middle grades probably induces 
some confusion. To just what extent this is true we do not 
know but we are sure that it is true to a significant degree. 

Finally, we have the large class of words which — when a 
sufficient number of children are tested — do show an increase 
in correctness from grade to grade, but which do not advance 
in anything like a constant ratio to the advance from grade to 
grade in spelling ability. Even among the words chosen as most 
favorable this discrepancy may be seen. The word " sure " 
(Table III) is 47 per cent correct in the 3d grade. This gives 
it a rank of 6th among the 25 words of the First Preferred List 
for that grade. In the 4th grade it advances to 55 per cent 
correct, but this advance is not sufficient to maintain its position. 
It falls to a rank where it is tied with " whistle " for 1 ith and 
12th place — i.e., its rank is 11.5. In the 5th, 6th, 7th, and 8th 
grades its rank remains fairly constant with the 4th-grade rank. 
It is 10.5, 14.5, 10.5, and 12. 

If this sort of irregularity is found in a word chosen as among 
the most regular, it will easily be seen how much greater would 
be the irregularity of words which were rejected. 

§ 6. Examination of the First Preferred List 

To further establish the value of the lists, we may investigate 
the behavior of the words as between grades in each school 
and in all schools combined. We shall begin with the First 
Preferred List. This list was chosen in the first instance from 
the returns of School I. It was taken to be the 25 words of 
closest correlation between the grades of that school. It was 
then referred to the other schools and the correlations were 
worked out for them. 



Examination of the First Preferred List 



17 



The method used was that suggested by Spearman in his 
article: "' Foot-rule ' for Measuring Correlation " (Spearman, 
'06). This method is explained and criticised by Brown in 
"The Essentials of Mental Measurement," pp. 71-76 (Brown, 
'n), and in a more elementary way by Whipple in his " Man- 
ual," pp. 34 and 35. (Whipple, '10.) 

The formula is 

R=i 

1/6 (iV 2 — 1) 
where S(#) denotes the sum of the "gains" in rank (sum of 
positive differences) of the second series on the first, and 1/6 
(N 2 — 1 ) is the value of the sum of such gains which may be 
expected by chance. These i^-values were then expressed as 
r- values (Pearson Coefficients of Correlation) by means of a 
table of equivalents. This table (Whipple, '10, p. 36), has been 

7T 

worked out from Spearman's conversion formula r = sin (-R). 

2 
The method is illustrated for the 4th and 5th grades of School 
I in Table V. 

By the formula, R = 1 , # = .72. This is 

T /6(N 2 — 1) 
equivalent to an r-value of .90. The correlation is therefore 
very satisfactory. Its Probable Error is .026, which is so small 
in relation to the obtained correlation, that the latter has a very 
high degree of reliability. 

The correlation of each grade with every other grade for 
School I was as follows : 



Correlation of 4th grade with 5th grade, 



4th 


" 6th 


" .88 


4th 


" 7th 


" .88 


5th 


" 6th 


" .92 


5th 


" 7th 


" .93 


6th 


" 7th 


" .88 




Average. . . 


QO 




(P.E 


= .026) 



90 



H 



1 8 Spelling Ability — Its Measurement and Distribution 



TABLE V 

School I. Coefficients of Correlation for 4th and 5th 
Grades Derived. Foot-rule Method 



even . . . 
lesson . . 
only. . . . 
smoke. . 
front . . . 

sure. . . . 

pear 

bought . 
another, 
forty. . . 

pretty. . 
wear . . . 
button . 
minute . 
cousin. . 

nails . . . 
janitor . 
saucer. . 
stopping 
sword . . 

freeze . . 
touch . . 
whistle . 
carriage . 
nor .... 



4th Grade 



Rank 



90 
80 

77 
77 
70 

70 
69 
67 
65 
63 

63 
56 
50 
49 
42 

42 
40 
40 
30 
25 

21 

21 
21 
17 
13 



5.5 
7 
8 
9 
10.5 

10.5 

12 

13 

14 

15.5 

15.5 
17.5 
17.5 
19 

20 

22 
22 
22 
24 
25 



5th Grade 



% 



Rank 



94 
85 
96 
81 
75 

85 
77 
83 
79 

72 

89 
73 
70 
65 
65 

79 
56 
65 
47 
50 

63 
52 
55 

45 
43 



2 

4.5 
1 
7 
11 

4.5 
10 

6 

8.5 
13 

3 

12 
14 
16 
16 

8.5 
19 
16 
23 
22 

18 
21 
20 
24 
25 



Gains, 
5th on 4th 



1 

2.5 

3.5 
5.5 



3 
2.*5 



1.5 

i" 
2 



29 = 2(g) 



It may be interesting before taking up the results in other 
schools to see to what extent the ' foot-rule ' method justifies 

itself in this kind of work. The 'product-moment' (r— ) 

ncr x <T 2 

and the 'unlike signs' methods (r = cos ^v) were used as a 
check. Table VI shows the result. 

It is evident from Table VI that if the true numerical state- 
ment of the amount of correlation may be expected to be at 
or near the average of all three methods, the one which tends 
most nearly to approximate the true result is here the ' foot-rule ' 



Examination of the First Preferred List 



19 



TABLE VI 

r-VALUES Between Grades of School I as Found by Three Methods 



Pairs of Grades 


Foot-rule 


Product- 
moment 


Unlike 
signs 


Average 


4th with 5th 


.90 
.88 
.88 
.92 
.93 
.88 


.92 
.76 
.73 
.78 
.79 
.91 


.99 


.94 


4th " 6th 


.97 .87 


4th " 7th 


.81 
.99 
.88 
.93 


.81 


5th " 6th 


.90 


5th " 7th 

6th " 7th 


.87 
.91 






Averages 


.90 


.82 


.93 


.88 



method. It has therefore seemed justifiable in future computa- 
tions of this sort to save the threefold labor and to rely upon 
this method. 

It is to be expected that since the list of words was in the main 
selected from the results in School I, the correlation will prove 
to be higher in that school than in any other. Such is indeed 
the case as will be seen by an inspection of Table VII. It will 
be remembered that School I has no 8th grade and that the 
3d grade in that school was not tested. 

We find that the correlations in School II are considerably 
lower than in schools III, IV, and V, even showing an apparent 
inverse relation in one instance. Yet the average of the coeffi- 
cients for School II is .42, the P.E. of which is less than .11. 
A correlation is entitled to scientific consideration if it is more 
than twice as large as its probable error. This one is nearly 
four times its probable error, and may therefore be regarded 
as satisfactory. Still more so are the relations between grades 
in the other schools; while, for all schools combined, the coeffi- 
cients of grade-to-grade correlation range from .47 to .93 with 
an average of .j6, AD =.12. Since for these same values 
P.E. ranges from .10 to .02 with an average at .057, the 
reliability of these values is adequate. 

It appears therefore that this list of words possesses the 
advantage of maintaining practically the same order of difficulty 
throughout the grades from the 3d to the 8th. In any grade 
the hardest word, the easiest word, and the words which 
take rank between tend strongly to hold their positions in every 



20 Spelling Ability — Its Measurement and Distribution 

other grade. Our list then is to a marked degree independent of 
fluctuations between grades. 

But this might be true and still leave something to be desired. 
Schools differ in many respects — in quality of teaching and 
supervision, in preferred methods, in word lists studied, in the 
character of the children as to economic and racial condition. 
The schools which we have under consideration differ widely 
in all those respects. These variations in local conditions may 
very likely produce considerable variation in the quality of the 
spelling output. 

TABLE VII 

Coefficients of Correlation. Grade with Grade, and Each 

Grade with All Grades for Each School. First 

Preferred List 



School 

3d with 4th 

« " 5th 

" " 6th 

" " 7th 

" " 8th 

" " entire school 

4th " 5th 

" " 6th 

" " 7th 

" " 8th 

" " entire school 

5th " 6th 

" " 7th 

" " 8th 

" " entire school 

6th " 7th 

" " 8th 

" " entire school 

7th " 8th 

" " entire school 

8th " entire school 

Average, grade with grade. 

Average, grade with school 



II 


III 


IV 


V 


.37 


.78 


.67 


.69 


.25 


.55 


.31 


.71 


— .07 


.40 


.45 


.75 


.40 


.23 


.34 


.75 


.01 


.31 


.47 


.67 


■35 


■79 


■77 


85 


.81 


.59 


.61 


.89 


.54 


.59 


.62 


.84 


.37 


.37 


.60 


.73 


.20 


.34 


.62 


.77 


•78 


.84 


.84 


92 


.72 


.60 


.90 


.93 


.60 


.69 


.88 


.90 


.52 


.48 


.83 


.72 


■9i 


.84 


.82 


■95 


.56 


.66 


.94 


.89 


.62 


.60 


.85 


.89 


■77 


■77 


.84 


■93 


.45 


.76 


.90 


.80 


•75 


.61 


•78 


.89 


.60 


■57 


.82 


.81 


.42 


.53 


.67 


.80 


.69 


■74 


.81 


.89 



All schools 



.79 
.71 
.55 
.47 
.71 
.82* 

.83 
.69 
.62 

.72 
,90* 

.90 
.86 
.93 

.98* 

.90 
,89 
.88* 



.86* 
.91* 
.76 
8n* 



* These r-values are for each grade with the grades of all schools 



Examination of the First Preferred List 



21 



TABLE VIII 

Correlations of School with School and of Each School with 
All Schools for Each Grade. First Preferred List 

School I is not included because of its different system of grading 



Grades 

School II with III 

II " IV 

II " V 

II " All 

" III " IV 

" III " V 

" III " All 

IV " V 

IV " All 

V " All 

Average, school with school 

Average, each school with all 
schools 



3d 

grade 



.45 
.32 
.61 
.62 

.76 
.54 
.88 

.77 
■83 

■83 

.58 



79 



4th 
grade 



5th 
grade 



6th 
grade 



■83 



7th 
grade 



62 
47 

77 

60 

83 

79 

56 



.82 



8th 
grade 



73 



All 
grades 



.70 
.80 
.88 
.pi* 

88 
81 
87* 

85 
93* 

93* 

82 



or- 



* These figures are for each school with all grades and schools combined, i.e., with all 
participants. 

A method in reading and word study which makes extensive 
use of the phonogram may possibly cause some words to become 
easy which are otherwise difficult. If the pupils in one school 
come from homes where English is not spoken, they may find 
difficult a set of words other than that which children of English- 
speaking parents find difficult. 

With the purpose of throwing some light on this point we 
shall consider what the correlation is between schools for each 
grade and for all grades with respect to the First Preferred List. 
Table VIII shows the correlation coefficients. The school-with- 
school average correlations range from .48 to .82 with a median 
at .69. The school-with-all-school averages range from .73 to 
.91 with a median at .83. A few of the coefficients throughout 
the table are low. There are, however, but six that are below 
.50. All but one of these are in the extreme grades (3d or 8th) 
and have to do with School II. The circumstances under which 
this school was examined account for this. The tests were given 



22 Spelling Ability — Its Measurement and Distribution 

immediately after the long summer vacation and the test-material 
comprised the Original List (270 words). The other schools 
were tested considerably later in the school year and the pupils 
in those schools wrote the Selected List (100 words). 

Notwithstanding these few shortcomings the 70 coefficients 
of Table VIII form an impressive argument for the value of the 
First Preferred List. We may fairly contend that not only 
are the positions of the words of this list relatively stable as 
between grades (Table VII), but that this permanency holds as 
between schools. 

§ 7. Examination of the Second Preferred List 

The second list of 25 words (see p. 13 or Appendix II) was 
examined in the same way that the first list was examined, i.e., 
with reference to correlations first between grades, and second 
between schools. 

At School I the correlations between grades were found to be 
as follows (Compare with similar tabulation for the First Pre- 
ferred List on page 17: 

4th grade with 5th grade .87 



4th 
4th 
5th 
5th 
6th 



c 6th " .83 
' 7th " .79 
1 6th " .95 
f 7th " .78 
' 7th « .83 

Average 84 

(P.E.=.o 4 ) 



For the other schools Table IX shows the correlations. It 
may be compared with Table VII (page 20). 

A comparison of Table IX with Table VII shows that 
although the word list to which Table IX refers was taken, on 
the basis of partial knowledge, to be somewhat inferior to the 
First Preferred List, these coefficients do not show it. The 
grade-with-grade averages (.66, .6y, .60, .48, and .75) are higher 
for some schools than in Table VII and lower for others. Their 
central tendency is almost identical while the closeness of group- 
ing is greater for the second than for the first list. Of the 



Examination of the Second Preferred List 



23 



105 measures of correlation in Table IX only 13 are less than 
four times their probable error, and only 3 are less than twice 
their probable error. The grade-to-grade relationships for all 
schools (column 6) range from .40 to .95, average .75, A.D.= 
.14. This is satisfactory to a degree scarcely, if at all, less than 
is the showing for the first list. 



TABLE IX 

Coefficients of Correlation. Grade with Grade and Each 

Grade with All Grades for Each School. Second 

Preferred List 



School 

3d with 4th 

" " 5th 

" " 6th 

" " 7th 

" " 8th 

" " entire school 

4th " 5th 

" " 6th 

" " 7th 

" « 8th 

" " entire school 

5th " 6th 

" " 7th 

" " 8th 

" " entire school 

6th " 7th 

" " 8th 

" " entire school 

7th " 8th 

" " entire school 

8th " entire school. 

Average, grade with grade. 

Average, grade with school 



II 


III 


IV 


V 


.60 


.75 


.55 


.38 


.69 


.62 


.56 


.26 


.61 


.55 


.55 


.11 


.43 


.55 


.41 


.07 


.51 


.52 


.35 


—.01 


■73 


.84 


-75 


■34 


.73 


.82 


.69 


.48 


.74 


.61 


.44 


.40 


.60 


.62 


.45 


.38 


.57 


.60 


.51 


.38 


.82 


.88 


■78 


.67 


.75 


.76 


.74 


.84 


.55 


.75 


.73 


.65 


.62 


.73 


.72 


.83 


.81 


.92 


.90 


.92 


.83 


.67 


.79 


.76 


.82 


.71 


.77 


.84 


■93 


.82 


■84 


.88 


.80 


.80 


.74 


.86 


.88 


•83 


.80 


■78 


.89 


.80 


.81 


.86 


.66 


.67 


.60 


.48 


.84 


■87 


.81 


■74 



All schools 

.74 
.73 
.55 
.40 
.41 
.72* 

.90 
.80 
.69 
.62 
■93* 

.90 
.83 
.80 
■95* 

.94 
.89 
.91* 

.90 
.87* 

.80* 

.75 

.86* 



* These r-values are for each grade with all grades of all schools. 

Table X (whose counterpart for the first list is Table VIII) 
reveals, as Table IX did not, the relative inferiority of the 
second list. There are 49 coefficients in Table X that are lower 
than the corresponding figures in Table VIII. Only 21 are 



24 Spelling Ability — Its Measurement and Distribution 



higher. The school-with-school averages are lower in five 
instances and higher in but two. The order of difficulty of 
these words is therefore not so permanent as between schools. 
It is, however, sufficient abundantly to justify the list. There 
are but six of the 70 coefficients in the body of the table that 
are less than four times their probable error, and but three that 
are less than twice their probable error. In some respects, 
indeed, this list is superior to the first list. A comparison of 

TABLE X 

Correlations of School with School and of Each School with 

All Schools for Each Grade. Second Preferred List 

School I not included. Compare with Table VIII, p. 21 



School II with III. 
II " IV. 
II " V. 
II " All. 



Ill 
III 
III 

IV 
IV 



IV. 

V. 

All. 

V. 
All. 



V " All 

Average, school with school 

Average, each school with all 
schools 



3d 

grade 



48 
29 
05 

57 

55 

47 
76 

47 

79 

62 
39 

69 



4th 
grade 



51 
17 
08 

56 

38 
43 

78 

43 

76 

57 
31 

68 



5th 
grade 



47 
52 
52 

84 

82 
62 

78 

55 

74 

82 
58 

,80 



6th 
grade 



79 



7th 
grade 



.60 
.45 
.43 
•76 

.61 
.72 

■85 

.81 
.65 

.84 
.60 

■78 



8th 
grade 



AU 

grades 



66 
43 
60 

84* 

59 

87 
88* 

65 

77* 



63 

82* 



* These figures are for each school with all grades and schools combined, i.e., with all 
participants. 

Tables III and IV shows that the words of the second list are 
in general more difficult than those of the first. Doubtless the 
first list is a somewhat better test for lower grades, while the 
second is a better test for upper grades. This supposition is 
neatly supported by the figures in Table X. Sixteen of the 21 
that are higher in this table than in Table VIII are in the three 
upper grades, while the two higher average correlations are in 
the 7th and 8th grades. If, therefore, some of the words in the 
first list are found to be too easy for the highest grades — as 



Conclusions Regarding the Preferred Lists 25 

doubtless they may be — then the second list will supply a valuable 
supplement to the first. 

§ 8. Conclusions Regarding the Preferred Lists 

Our lists therefore prove to be well selected. Success and 
failure in spelling them may be used with considerable confidence 
to measure the thing we call spelling ability. The establishment 
of this fact is of the utmost importance. In general when we 
are to measure mental traits or capacities the thing we directly 
measure is itself a physical phenomenon or fact. We measure 
fatigue by the number and height of lifts with the ergograph, or 
by the distance between points of the esthesiometer necessary to 
be recognized as * two.' We measure attention and perception 
by counting dots or by cancellation ; memory, by the number of 
digits reproduced; association by the number of words pro- 
nounced in a given time; and intelligence itself, by a series of 
tests which may be scored objectively. What we deal with 
directly is something, assumed to be functionally related to the 
trait in question, which can be measured in time or space or 
which can be counted. If this objective manifestation does not 
accurately register the subjective state to which it is supposed 
to correspond, it is impaired, to the extent of its inaccuracy, as 
an index to be directly measured. 

Now it is undoubtedly true that the misspellings of most words 
are unreliable as indicating lack of spelling ability in general; 
and on the other hand it is probable that to spell them correctly 
often argues little more than that the subject can spell the 
particular words that he did spell. Most words are in some way 
special — and they are special (particularly for children) in ways 
that we do not realize. Very often they do not mean the same 
thing to one person that they do to another. They are frequently 
pronounced differently by different people. They suggest dis- 
similar imagery. They connote variously. They range from 
very easy to very hard ; and those that are easy for some people 
are hard for others. Moreover there are numerous ways of 
misspelling them, each indicating its own causal incoordination. 
An error may not be equal to an error even in misspelling the 
same word. 



26 Spelling Ability — Its Measurement and Distribution 

It would be presumptuous to suppose that all these difficulties 
have been overcome in selecting our two preferred lists. Without 
doubt we have only roughly approximated the ideal conditions 
under which a physical fact may be the transcript of a mental 
trait. Probably nothing more satisfactory than an approxima- 
tion can be devised. But we have been at no small pains to 
secure a list of words which would be free from many of these 
variations, and we think we have done so. From an inspection 
of them we may be justified in believing that their pronunciation 
and meaning are fairly constant for everybody; and we may 
regard it as probable that their associative connections do not 
vary much for different people. From a statistical analysis of 
them we find that their behavior with elementary school children 
is constant to a marked degree, and in particular that it is relative- 
ly independent of maturity and of local conditions. Older 
children in higher grades spell them more frequently and in each 
grade more frequently than in the one before it. Children in 
schools under favorable circumstances do better with them than 
do children in less favorable .situations. It is because they re- 
flect these conditions that they are valuable. By the use of 
them, conditions in a school, a class, or an individual may be 
revealed ; and conversely to a certain extent if the conditions are 
known (e.g., the grade) the results, by the use of them, are pre- 
dictable. 

These lists, then, tend strongly to* remain intact under various 
conditions. As lists they appear to be reliable, and our numerical 
results give quantitative expression to this reliability. But as 
to the words themselves, we shall yet have much to say. There 
has been no attempt to secure lists composed of words of equal 
difficulty. The effort has rather been to choose words which 
differ widely in this respect. We shall now attempt to arrange 
these words on a scale which shall accurately represent their 
difficulty, assuming as true a certain supposition concerning the 
form of distribution of spelling ability within a school grade. 
The resulting scale will represent their difficulty approximately 
in so far as this supposition is approximately true. When this 
is done, their value for test purposes independently of the list 
which contains them will be established. 



Ratings of Individual Pupils 



27 



§ 9. Ratings of Individual Pupils 

In addition to scoring words, the papers of individual pupils 
were rated. This was done in the usual manner, the ability to 
spell one word being scored as equal to the ability to spell any 
other word of the list. This procedure is subject to the criticism 
made in Section 1 above; but in the absence of any evaluation 
of the words, a system of weighting is not possible. The results 
will not be misused here. 

The test material consisted of the 100 words of the Selected 
List. The papers written at schools II, III, IV, and V were 
used. School I was not available because of its system of grad- 
ing. A few papers were incomplete in each of the schools ; these 
were rejected in this part of the work. In all, 2,487 papers were 
rated. Table XI shows for each grade the distribution of in- 



TABLE XI 

Distribution of Individual Ratings of Pupils in 
Schools II, III, IV and V 



Per- 
centage 


3d Grade 


4th Grade 


5th Grade 


6th Grade 


7th Grade 


8th Grade 




















Correct 


No. 


% 


No. 


% 


No. 


% 


No. 


% 


No. 


% 


No. 


% 


0- 5 


9 


2.0 


1 


.2 


1 


.2 














6- 10 


22 


4.9 


1 


.2 


2 


.4 














11- 15 


30 


6.7 


10 


2.1 


1 


.2 














16- 20 


38 


8.5 


12 


2.6 


2 


.4 














21- 25 


44 


9.9 


13 


2.8 


6 


1.2 


2 


.5 










26- 30 


47 


10.6 


23 


4.9 


12 


2.3 














31- 35 


34 


7.6 


29 


6.2 


13 


2.5 


2 


.5 


2 


.5 






36- 40 


38 


8.5 


27 


5.8 


11 


2.1 














41- 45 


24 


5.4 


30 


6.4 


18 


3.5 


6 


1.4 


2 


.5 






46- 50 


34 


7.6 


33 


7.1 


28 


5.4 


4 


1.0 


1 


.3 






51- 55 


26 


5.8 


27 


5.8 


20 


3.9 


6 


1.4 


3 


.8 






56- 60 


24 


5.4 


31 


6.6 


32 


6.2 


15 


3.6 


5 


1.4 


1 


.4 


61- 65 


26 


5.8 


39 


8.4 


44 


8.5 


12 


2.9 


6 


1.6 


1 


.4 


66- 70 


17 


3.8 


29 


6.2 


48 


9.3 


23 


5.5 


8 


2.2 


3 


1.1 


71- 75 


13 


2.9 


45 


9.6 


49 


9.5 


30 


7.2 


18 


4.9 


8 


2.9 


76- 80 


8 


1.8 


35 


7.5 


59 


11.5 


52 


12.4 


31 


8.5 


11 


4.0 


81- 85 


4 


.9 


33 


7.1 


37 


7.2 


67 


16.0 


38 


10.4 


19 


6.9 


86- 90 


4 


.9 


26 


5.6 


64 


12.4 


61 


14.6 


79 


21.6 


41 


14.8 


91- 95 


3 


.7 


19 


4.1 


50 


9.7 


101 


24.2 


93 


25.5 


80 


28.9 


96-100 






4 


.9 


18 


3.5 


37 


8.9 


79 


21.6 


113 


40.8 


Totals... 


445 




467 




515 




418 




365 




277 




Medians. 




35.8 




60.70 




73.10 




84.90 




90.50 




94.68 


A. D.... 




18.0 




20.9 




10.4 




10.0 




7.9 




5.8 



28 Spelling Ability — Its Measurement and Distribution 



dividual ratings. It reads as follows : "In the 3d grade 9 pupils 
were rated between 0% and 5%, which was 2.0% of all the 3d 
grade pupils. In the 4th grade 1 pupil was rated between 0% 
and 5%, which was .2% of all the 4th grade pupils," etc. 

The striking characteristic of the distribution of these ratings 
is their extreme variability. Children of the 3d grade are repre- 
sented in every group but one, while children of the 4th and 5th 
grades are rated in every group. It appears that we may expect 
a few 6th- and 7th-grade children to spell not more than 20 or 
30 of these hundred words, which is not quite as good as the 
typical ability of 3d-grade children for the same words. The 
8th-grade pupils show the least variation. This is probably 
true of this grade in general. It is not, however, as marked as 
these figures indicate. The 100-word list as a whole, whatever 
may be true about some of the individual words, did not thorough- 
ly test this grade. A glance at Fig. 6 will show how sharply cut 
off at the high end is the curve of distribution. This is against 
all the facts which we know about eighth-graders in particular 
and human ability in general. A harder test would have shown 
a lower mode and a more gradual tapering off at the upper end 
of the curve. But even as this record stands we may look for 
a considerable number — between 7 and 8 per cent — of 8th-grade 
pupils to average no better than the typical performance of 5th- 
grade children. 

TABLE- XII 
Distribution of Individual Ratings Grouped to Show Modes. Figs. 1-7 



Percentage 
Correct 


3d Grade 


4th Grade 


5th Grade 


6th Grade 


7th Grade 


8th Grade 


0-10 


6.91 

22.1 
15. 2J 

20.5) 

36.6 
16. lj 

13.01 

24.2 
11.2J 

9.61 

14.3 

4.7J 

1.81 

A 2 - 6 


• 4 | 5.X 

4.7J 

7.71 

19.7 
12. OJ 

13.51 

25.9 
12. 4J 

14.61 

31.7 

17. IJ 

12.71 

17.7 
5.0J 


.61 

1.2 

• 6J 

3.51 

8.1 
4.6J 

8.91 

19.0 
10. IJ 

17.81 

\ 38.8 
21. 0J 

19.61 

\ 32.8 
13. 2J 


:3 " 

2.41 

7.4 
5.0J 

8.41 

28.0 
19.6J 

30.61 

63.7 
33. IJ 


.5j .5 

.81 

3.0 
2.2J 

3.81 

\ 17.2 
13. 4J 

32.01 

79.1 

47.1 J 




11- 20 




21- 30 




31- 40 




41- 50 


,} • 


51- 60 


61- 70 


1.51 

8.4 
6.9J 


71- 80 


81- 90 


21.71 

[ 91.4 
69. 7J 


91-100 







Ratings of Individual Pupils 



29 





JO 2.0 30 40 SO to 70 SO '90-/00 

& 9 .3. 



10 10 30 40 50 60 70 8° <?0 WO 

%1 



O /O *0 <30 #) o~o <So 70 so 90 /oo 

H 3 A 




O iO 20 30 40 & Co 70 80 90 loo 

Fig. 5. 



Fig. 1 
Fig. 2 
Fig. 3 
Fig. 4, 
Fig. 5 



Frequency of each rating (of per cent of Selected List spelled 

correctly) in grade 3. 
Frequency of each rating (of per cent of Selected List spelled 

correctly) in grade 4. 
Frequency of each rating (of per cent of Selected List spelled 

correctly) in grade 5. 
Frequency of each rating (of per cent of Selected List spelled 

correctly) in grade 6. 
Frequency of each rating (of per cent of Selected List spelled 

correctly) in grade 7. 



3° 


Spelling Ability — Its Measurement and Distribution 




70 








C5 








CO 








SS 








60 








45 








40 








3S 




Fig. 6 




30 








2S 
20 


J 








IS 








10 








5 








10 20 30 ft) 50 CO 70 SO 90 too 


2$- 




*o 










/& 








/C 


|J 




Fig. 7 


S 


r s T ^^ 





/o Zg so y so £0 70 80 fo JOo 



Fig. 6. Frequency of each rating (of per cent of Selected List spelled 

correctly) in grade 8. 

Fig. 7. Frequency of each rating (of per cent of Selected List spelled 

correctly) in grades 3-8 combined. 



Overlapping 3 1 

Another characteristic of the distributions shown in Table XI 
is the absence of clearly marked modes. Table XII is a group- 
ing of the per cent columns of Table XI into io's and 20's. From 
this grouping wide modes of marked character are shown. 

Figs. 1 to 7 show the same facts graphically. From the nature 
of these figures the test material appears to have been capable 
of revealing satisfactorily the spelling ability of grades 3, 4, and 
5. Figs. 1 to 7 are the surfaces of frequency of spelling ability 
with the Selected List. In each of them the horizontal scale 
shows percentages correct; the vertical scale shows the per cent 
of children having each rating for percentage correct, by steps 
of 10. The number of children represented is 445 in grade 3, 
467 in grade 4, 515 in grade 5, 418 in grade 6, 365 in grade 7, 
and 2JJ in grade 8. 

§ 10. Overlapping 

It follows as a matter of course from the variability of these 
ratings that the overlapping of grade on grade is conspicuous. 
We have located the median abilities, of each grade, for the 
selected word list. They are: 3d grade, 35.8; 4th grade, 60.7; 
5th grade, 73.1 ; 6th grade, 84.9; 7th grade, 90.5; 8th grade, 94.7 
(See Table XI). Table XIII shows the number of pupils and 
the per cent of pupils in each grade whose ratings equalled or 
exceeded the medians of every other grade. The table reads 
as follows : In the 3d grade j6 pupils equalled or exceeded the 
median rating of the 4th grade which was 17.1% of all the 3d- 
grade pupils ; 2J equalled or exceeded the median rating of the 
5th grade which was 6.1% of all the 3d-grade pupils. In the 4th 
grade 378 pupils equalled or exceeded the median rating of the 
3d grade which was 80.9% of all the pupils of the 4th grade, 
etc. It will be noticed that there are two places where there is 
no overlapping. There are no 3d-grade children who equal the 
median rating of the 8th grade, although the 3 who exceed the 
7th-grade median come very near it. Two of them are rated at 
93 and one at 94, only 1.7 and .7 below the 8th-grade median. 
There is also no overlapping of the 8th grade on the 3d. All the 
pupils of the 8th grade exceed the median of the 3d grade. When, 
however, we say that at these points there is no overlapping, 
we do not mean that their surfaces of frequency do not enclose 



32 Spelling Ability — Its Measurement and Distribution 



TABLE XIII 

Number and Per Cent of Pupils in Each Grade Whose Ability 

Equalled or Exceeded that of the Median Pupil 

in Every Other Grade 



3d grade. . . 

N=445. . . . 
Med.=35.8 

4th grade. . 
N=467.... 
Med.=60.7 

5th grade. . 
N=515. . . . 
Med.=73.1 

6th grade . . 
N=418.... 
Med.=84.9 

7th grade . . 
N=365. . . . 
Med.=90.5 

8th grade. . 
N=277. . . . 
Med.=94.7 





3d 
Grade 


4th 
Grade 


5th 
Grade 


6th 
Grade 


7th 
Grade 


No. 

% 




76 
17.1 


27 
6.1 


9 
2.0 


3 
0.7 


No. 

% 


378 
80.9 




146 
31.3 


52 
11.1 


27 
5.8 


No. 

% 


478 
92.8 


370 

71.8 




142 
27.6 


73 

14.2 


No. 
% 


414 
99.0 


384 
91.9 


338 
80.1 




142 
34.0 


No. 

% 


363 
99.5 


354 
96.4 


328 
89.9 


256 
70.1 




No. 

% 


277 
100 


276 
99.6 


269 
97.1 


241 
87.0 


200 
72.2 



8th 
Grade 



9 
1.9 



30 

5.8 



57 
13.6 



99 
27.1 



common areas. If Fig. I is placed on Fig. 8 so that the zero 
points coincide, it is evident that there is considerable area com- 
mon to both. We mean that the upper part of the 3d-grade 
surface of frequency does not lap over the median point of the 
8th-grade surface, and that the lower part of the 8th-grade sur- 
face does not reach down to the 3d-grade median. There are 
many 3d-grade children who do better than the poorest 8th- 
grade children. 

The fact is, then, that except as between the 3d and 8th grades, 
some pupils of each grade perform like typical children of every 
other grade. Since this is true, it serves to fix the location of 
the frequency curves and medians for each grade with reference 
to each other. For the purpose of doing so we shall for the pre- 
sent assume that the distribution of spelling ability in each grade 
is "normal," i.e., is correctly represented by the curve of error. 

There is some argument for this assumption. The fact that 



Overlapping 



33 



our surfaces of frequency (Figs. 1-6) do not, especially for upper 
grades, closely resemble the normal curve, only shows that the 
test material was not difficult enough to bring out a distribution 
in real accordance with spelling ability. The result of using a 
different list of words is shown for grades 6, 7, and 8 by Figs. 
8, 9, and 10. The test material in this instance was Rice's "Sen- 
tence Test": 396 children in the 6th grade, 367 in the 7th, and 
244 in the 8th wrote this test in schools II, III, and VIII. The 




O 10 20 60 qo SO 60 70 SO <i0 JQO 

FTq 8 




3S 
30 
26 
20 
IS 
/o 

5 



10 2.0 30 40 SO CO 70 80 90 /oo 

Fig.<? 



A) 20 30 *0 SO CO 70 80 90 100 

Fig.lft 



Figs. 8, 9 and 10. Frequency of each rating (of per cent of Rice Sen- 
tence List spelled correctly) in grades 6, 7 and 8, respectively. 
AT==396 for grade 6; 367 for grade 7; and 244 for grade 8. The 
horizontal scale is for per cent spelled correctly; the vertical scale 
is for the percentage of children receiving each rating for percentage 
correct, by steps of 10. 



surface of frequency for the 6th grade is close to the "normal" 
surface. If that for the 7th and 8th grades is less so, it is still 
far more regular than the surfaces shown for these grades in 
Figs. 5 and 6 and might be made still more so by an appropriate 
selection of test material. There seems no good ground for as- 



34 Spelling Ability — Its Measurement and Distribution 

suming that the distribution of spelling ability in any grade is 
not according to the normal curve or according to a curve which 
resembles it closely. 

§ ii. Location of Grade Medians 
Upon the assumption, therefore, of a normal distribution we 
may proceed to locate the grade medians with reference to 
each other. In all cases we shall work with per cents instead of 
with numbers of pupils. This will reduce all surfaces of fre- 
quency to equal areas. We shall assume further that the real 
variability of any one of these grades in spelling ability is equal 
to the real variability of any other one of them. 

We have already seen (Table XIII) that 17.1% of the 3d- 




A 
Fig. 11 



Showing the overlapping of the 3d and 4th grade surfaces of 
frequency. 



grade pupils equal or exceed the median ability of 4th-grade 
children. Fig. 11 shows this fact by a diagram. The surface 
on the left (Axis OM) represents the 3d-grade distributions. M 
is its median point, MD (=MQ) is its probable error — i.e., 
figure NPQD is one-half its area, thus graphically representing 
one-half the cases in the 3d grade, which accordingly do not 
deviate from the median by an amount greater than MD. The 
surface on the right represents the 4th-grade distribution beyond 
whose median axis, 1 M 1 , the 3d-grade surface extends to an 
amount represented by the shaded figure KCM 1 . This stands 
for the 3d-grade children who equal or exceed the 4th-grade 
median — i.e., it is 17.1% of the 3d-grade surface of frequency. 
Accordingly the area OKM 1 M represents 32.9% of the cases. 
The distance MM 1 may now be found in terms of P.E. It is 
the distance from the median point along the X-axis of the normal 
probability integral which includes 32.9% of the cases. Distances 



Location of Grade Medians 



35 



corresponding to different per cents of the total area of the 
curve have been worked out. Table XIV, which is taken from 

TABLE XIV 

Table of Values of the Normal Probability Integral Corre- 
sponding to Values of P.E. Total Area of the 
Surface of Frequency Taken as 10,000 



X 






X 






X 






X 









No. 


A 





No. 


A 





No. 


A 





Xo. 


A 


P.E. 


Cases 




P.E. 


Cases 




P.E. 


Cases 




P.E. 


Cases 














81 






18 














135 


1.5 


3441 


80 


3.00 


4785 


17 


4.5 


4988 




.05 


135 


134 


1.55 


3521 


76 


3.05 


4802 


15 


4.55 


49S9 




.1 


269 


134 


1.6 


3597 


74 


3.1 


4817 


14 


4.6 


4990 




.15 


403 


133 


1.65 


3671 


71 


3.15 


4831 


14 


4.65 


4991 




.2 


536 


134 


1.7 


3742 


69 


3.2 


4845 


13 


4.7 


4992 




.25 


670 


132 


1.75 


3811 


65 


3.25 


4858 


12 


4.75 


4993 




.3 


802 


131 


1.8 


3876 


63 


3.3 


4870 


11 


4.8 


4994 


.6 


.35 


933 


130 


1.85 


3939 


61 


3.35 


4881 


10 


4.85 


4994.6 


.6 


.4 


1063 


130 


1.9 


4000 


57 


3.4 


4891 


9 


4.9 


4995.2 


.5 


.45 


1193 


128 


1.95 


4057 


56 


3.45 


4900 


9 


4.95 


4995.7 


.5 


.5 


1321 


126 


2.0 


4113 


53 


3.5 


4909 


8 


5.0 


4996.2 


.4 


.55 


1447 


124 


2.05 


4166 


51 


3.55 


4917 


7 


5.05 


4996.6 


.5 


.6 


1571 


124 


2.1 


4217 


48 


3.6 


4924 


7 


5.1 


4997.1 


.3 


.65 


1695 


121 


2.15 


4265 


46 


3.65 


4931 


6 


5. 15 


4997.4 


.3 


.7 


1816 


119 


2.2 


4311 


43 


3.7 


4937 


6 


5.2 


4997.7 


.3 


.75 


1935 


118 


2.25 


4354 


42 


3.75 


4943 


5 


5.25 


499S.0 


.2 


.8 


2053 


115 


2.3 


4396 


39 


3.8 


4948 


5 


5.3 


4998.2 


.2 


.85 


2168 


113 


2.35 


4435 


37 


3.85 


4953 


4 


5.35 


4998.4 


.2 


.9 


2281 


111 


2.4 


4472 


36 


3.9 


4957 


4 


5.4 


4998.6 


.2 


.95 


2392 


108 


2.45 


4508 


33 


3.95 


4961 


4 y 


5.45 


4998.8 


.2 


1.0 


2500 


106 


2.5 


4541 


32 


4.0 


4965 


3 


5.5 


4999.0 




1.05 


2606 


103 


2.55 


4573 


29 


4.05 


4968 


3 


5.55 


4999.1 




1.1 


2709 


101 


2.6 


4602 


29 


4.1 


4971 


3 


5.6 


4999.2 




1.15 


2810 


98 


2.65 


4631 


26 


4.15 


4974 


3 


5.65 


4999.3 




1.2 


2908 


96 


2.7 


4657 


25 


4.2 


4977 


2 


5.7 


4999.4 




1.25 


3004 


93 


2.75 


4682 


23 


4.25 


4979 


2 


5.75 


4999.5 


.05 


1.3 


3097 


91 


2.8 


4705 


22 


4.3 


4981 


2 


5.8 


4999.55 


.05 


1.35 


3188 


87 


2.85 


4727 


21 


4.35 


4983 


2 


5.85 


4999.6 


.05 


1.4 


3275 


85 


2.9 


4748 


19 


4.4 


4985 


2 


5.9 


4999.65 


.05 


1.45 


3360 




2.95 


4767 




4.45 


4987 




5.95 


4999.7 





36 Spelling Ability — Its Measurement and Distribution 

Thorndike ('13', p. 200), presents these distances in units of P.E. 
By reference to it we find that 32.9% corresponds to 1.4088 P.E. 
In a similar way, the 6.1% of 3d-grade children who equal 
or exceed the 5th-grade median (Table XIII) serve to locate 
that median at 2.2929 P.E. above the 3d-grade median. The 6th- 
grade median is 3.0441 P.E. and the 7th, 3.6429 P.E. above the 
3d-grade median. The distance between the 3d- and 8th-grade 
medians cannot be directly calculated owing to the absence of 
sufficient overlapping. 



M: 



AJ ¥ M* ^Al, At* 



B 



Suppose the line AB to represent the range of spelling ability 
in the elementary school. At a certain distance above A, the 
absolute zero-point, stands the 3d-grade median, M s . Above it 
and at distances to be determined are the medians of the 4th 

to the 8th grades, M 4 M 8 . For brevity we shall call the 

distance from the 3d- to the 4th-grade median M 3 _ 4 , etc. M 4 _ 3 
means the same distance as M 3 _ 4 , but measured in the opposite 
or negative direction. 

TABLE XV 

The Per Cent of Pupils in Each Grade Whose Ability Equalled 
or Exceeded that of the Median Pupil in Every Other 
Grade; with the P.E. Values Correspond- 
ing to Each Per Cent 







3d 
Grade 


4th 

Grade 


5th 
Grade 


6th 
Grade 


7th 
Grade 


8th 
Grade 


3d grade. . . . 


% 
P.E. 




17.1 
1 . 4088 


6.1 

2.2929 


2.0 
3.0441 


0.7 
3.6429 



? 


4th grade . . . 


% 
P.E. 


80.9 
—1.2962 




31.3 

.7227 


11.1 

1.8111 


5.8 
2.3308 


1.9 
3.0767 


5th grade . . 


% 
P.E. 


92.8 
—2.1663 


71.8 
—.8553 




27.6 
.8819 


14.2 

1.5888 


5.8 
2.3308 


6th grade . . . 


% 
P.E. 


99.0 
—3.4500 


91.9 
—2.0735 


80.1 
-^-1.2532 




34.0 
.6117 


13.6 
1.6291 


7th grade . . 


% 
P.E. 


99.5 
—3.8200 


96.4 
—2.6673 


89.9 
—1.8918 


70.1 

— .7818 




27.1 
.9041 


8th grade . . 


% 
P.E. 


100 


99.6 
—3.9375 


97.1 
—2.8114 


87.0 
—1.6704 


72.2 
—.8730 





Location of Grade Medians 3 7 

Table XV gives all the distances between medians which our 
data permit us to calculate directly. The P.E. values, reading 
across the table, indicate that on the record of the pupils tested 
the 4th-grade median is found to be 1.4088 P.E. above the 3d- 
grade median, the 5th 2.2929 P.E. above it, the 6th 3.0441 P.E. 
above it, and the 7th 3.6429 P.E. above it; that the 3d-grade 
median is 1.2962 P.E. below the 4th-grade median, the 5th .7227 
P.E. above it, etc. 

It will be seen that i¥ 4 is given as 1.4088 above M s , while M 3 
is given as only 1.2962 below ikf 4 . We shall have to adopt one 
or the other, or some value between them as the most probably 
correct distance, M z _±. Similarly for each of the other distances 
(except M 3 - 8 ) we have two values, and these two values are in 
each case somewhat different one from the other. The following 
are the pairs of values which Table XV shows : 



^3-4 


1.4088 and 1.2962 


^3-5 


2.2929 


u 


2.1663 


M 3-6 


3.0441 


u 


3.4500 


M 3 _ 7 


3.6429 


u 


3.8200 


■^4-5 


.7227 


" 


.8553 


^4-6 


1.8111 


u 


2.0735 


M 4 _ 7 


2.3308 


u 


2.6673 


^4-8 


3.0767 


u 


3.9375 


^5-6 


.8819 


u 


1.2532 


^5-7 


1.5888 


« 


1.8918 


^5-8 


2.3308 


u 


2.8114 


^6-7 


.6117 


u 


.7818 


M 6-8 


1.6291 


u 


1.6704 


M 7 _s 


.9041 


u 


.8730 



The differences between these pairs of values is in most 
cases small. In all cases they afford data for the determination 
of the distances between medians which will be probably more 
accurate than either of them. 

We do not, however, need all these values. If we have five, 
namely, M 3 _ 4 , M 4 _ 5 , M 5 ^, M 6 . 7 , and M 7 . 8 , all the others may 
be obtained by adding these together. We shall therefore 
attempt to derive as accurately as possible these five values in 
terms of the unit, P.E. Each of them is represented directly by 
two quantities as shown above. But it is clear that if we use 
more of the data of Table XV we may obtain values whose 



38 Spelling Ability — Its Measurement and Distribution 

accuracy will be much more satisfactory. We may, for instance, 
find for the distance between the 4th-grade median and the 
5th-grade median (M 4 _ 5 ) a third value by subtracting from the 
distance between the 3d- and 5th-grade medians (M 3 _ 5 = 2.2929) 
the distance between the 3d- and 4th-grade medians (M 3 . 4 = 
1.4088). This gives .8841. Another value is the difference 
between the same two distances expressed negatively, i.e., accord- 
ing to our notation, between Af 5 _ 3 (2.1663) and M 4 _ 3 (1.2962;, 
which is .8701. Again we may use the difference between Af 4 _g 
and ikf 5 -6, between M^ and M 5 . 7 , between M 4 _ 8 and M 5 _ 8 ; 
and for each of these differences between positive quantities 
we have a difference between corresponding negative quantities. 
This adds six more expressions, making ten altogether, for the 
same distance, M 4 _ 5 . This is of course only a beginning of the 
great number of combinations which may be used to get expres- 
sions for the same distance. We think these few, however, 
since they use each of the 18 segments (nine counted both ways) 
which terminate at either M 4 or M 5 will be sufficient to determine 
M 4 _ 5 with considerable accuracy. We doubt whether the remoter 
segments (e.g., M 6 _ 7 , M 6 . 8 , and M 7 _ 8 ) would, if used, increase 
the accuracy at all. 

Accordingly we have calculated 10 values for M^ 5 , M 5 ^, and 
ikf 6 _ 7 . Since we have no expression of direct relation between 
M 3 and M 8 , we have but 8 values for ikf 3 _ 4 and M 7 _ 8 . Table 
XVI gives all these values with the derivation of each. It also 
gives the averages, unweighted and weighted, of the values for 
each of the five median intervals. 

It was felt that to give each of these items the same weight 
was to fail to take account of their reliability. The direct values 
are, no doubt, most to be depended upon. Those computed by 
using a distance which passes over one median are less so. Those 
involving two or more of these " skips " are still less so and 
diminish in reliability as the number of " skips " increases. It 
will be found that in column 2 of Table XVI the first quantity 
.8841 is derived by using a value that involves one skip. ikf 3 _ 5 
skips over M 4 , while M 3 _ 4 , which is taken from it, presents no 
skips. The second quantity, .7227, is direct, and there are no 
skips. .9292 involves one skip, .7420 three skips (M 4 . 7 skips M 5 
and M 6 , and M 5 _ 7 skips M 6 ), etc. It will be found upon trial 



Location of Grade Medians 

TABLE XVI 

Direct and Derived Values op Median Distances in Terms of P.E. 



39 



/3AV 



o.zsi ... tost . cUt r <>*><> 



^ 



tfyr 



/* til 



^s 



B 





^3-4 


^4-5 


^5-0 


^6-7 


M 7 _ 8 




1 . 4088 
(direct) 

1.5704 

(M 3 _-M 4 _ 5 ) 

1.2330 
(M 3 _ 6 ~M 4 _ 6 ) 

1.3121 

(M 3 _ 7 -M 4 _ 7 ) 

? 

(M 3 _ 8 — M 4 _ 8 ) 

1.2962 
(direct) 

1.3110 

(M 5 _ 3 -M 5 _ 4 ) 

1 . 3765 
(M 6 _—M^) 

1.1527 

(M 7 _ 3 -M 7 _ 4 ) 

? 

(M 8 _ 3 — M 8 _ 4 ) 


.8841 
(M 3 _ 5 -M 3 _ 4 ) 

.7227 
(direct) 

.9292 
(M 4 _ 6 -M 5 _ 6 ) 

.7420 
(M 4 _-M 5 _ 7 ) 

.7459 

(M 4 _ 8 -M 5 _ 8 ) 

.8701 
(M 5 _ 3 -M 4 _ 3 ) 

.8553 
(direct) 

.8203 
(M 6 _-M 6 _ 5 ) 

.7755 

(M 7 _ 4 -M 7 _ 5 ) 

1.1261 

(M 8 _-M 8 _ 5 ) 


.7512 
(M 3 _ 6 -M 3 _ 5 ) 

1.0884 

(M 4 _ 6 -M 4 _ 5 ) 

.8819 
(direct) 

.9771 

(M 5 _— M 6 _ 7 ) 

.7017 

(M 5 _ 8 — M 6 _ 8 ) 

1.2837 
(M 6 _ 3 — M 5 _ 3 ) 

1.2182 

(M 6 _ 4 -M 5 _ 4 ) 

1 . 2532 

(direct) 

1.1100 

(M 7 _ 5 -M 7 _ 6 ) 

1.1410 

(M 8 _ 5 —M 8 _ 6 ) 


.5988 
(M 3 _ 7 -M 3 _ ) 

.5199 

(M 4 _-M 4 _ 6 ) 

.7069 
(M 5 _— M 5 _ 6 ) 

.6117 
(direct) 

.7250 

(M 6 _ 8 — M 7 _ 8 ) 

.3700 
(M 7 _ 3 — M 6 _ 3 ) 

.5938 
(M 7 _-M 6 _ 4 ) 

.6386 
(M 7 _ 5 -M 6 _ 5 ) 

.7818 
(direct) 

.7974 
(M 8 _ Q — M 8 _ 7 ) 


9 

(M 3 _ 8 — M 3 _ 7 ) 

.7459 
( M 4 _ 8 -ilf 4 _ 7 ) 

.7420 
(M 5 _ 8 -M 5 _ 7 ) 

1.0174 
(M 6 _ 8 — M 6 _ 7 ) 

.9041 
(direct) 

1.2702 

(M 8 _-M 7 _ 4 ) 

.9196 
(M 8 _ 5 -M 7 ^) 

.8886 
(M 8 _ 6 — M 7 _ 6 ) 

.8730 
(direct) 


Average 

Weighted 
Average 


1 . 3326 
1.3505 


.8471 
.8363 


1.0406 
1.0505 


.6344 
.6608 


.9201 
.9101 



that in all values there are either o, i, 3, or 5 skips. We have 
weighted them 6, 4, 3, and 2 respectively (ratio about 1.5). 
This is, of course, pure assumption, nor do we know of any 
convenient plan of weighting which would not be. All we can 



40 Spelling Ability — Its Measurement and Distribution 

say is that to us the direct values seem to be quite one and 
one-half times as reliable as those involving a distance which 
passes over one median, that it seems reasonable the latter 
should be about as many times more reliable than those involving 
3 skips, and that the derivation with 5 skips would be inferior 
in approximately the same ratio. Weighting therefore as above 
indicated, we obtain values for the median distances as given in 
the last line of Table XVI. These are the measures that will 
be used in this study ; but they differ so little from those obtained 
without weighting that the latter may serve almost as well. 

Concretely these results mean that if we represent the differ- 
ence between no spelling ability at all and the ability of typical 
3d-grade children by x, the ability of typical 4th-grade children 
will be represented by #+1.351, of typical 5th-grade children 
by x+ 1. 35 1 +.836 or x + 2.187, of typical 6th-grade children by 
x + 3.238, of typical 7th-grade children by x + 3.899, and of 
typical 8th-grade children by x + 4.809. (The determination 
of the value of x is not material in the present connection. We 
shall, however, have something to offer on this point in a later 
section.) The median distances suggest that so far as spelling 
is concerned the equal time intervals of one year between the 
grades do not at all correspond to the differences in ability. 
The difference between 3d-grade performance and 4th-grade 
performance is more than twice as great as the difference 
between 6th- and 7th-grade performance. Whether this is due 
to a more or less common failure in the 7th grade to give as 
much attention to spelling as in earlier grades or whether in 
general 6th and 7th grades are actually closer together than 
others, is a question which we cannot settle. That a lack of 
effort to instruct in spelling in the higher grades does not fully 
account for the differences is suggested by the fact that the 
8th grade stands at a greater distance from the 7th than the 5th 
does from the 4th or the 7th from the 6th. 

§ 12. Scaling the Words 

Assuming that the normal surface of frequency represents 
the distribution of spelling ability in each grade, we shall now 
seek to determine how difficult each one of the 50 words listed 



Scaling the Words 



4i 



in Tables III and IV is for each grade. A word spelled by one 
hundred per cent of the pupils in, say, the 3d grade would have 
no difficulty for that grade. The ability of all pupils would be 
greater than the ability required to spell it, and the entire area 
of the frequency surface would lie above it — i.e., to the right of 
it. In Fig. 12, if OP represents the Probable Error, it would be 
located theoretically at an indefinite distance to the left of the 
point 0, a distance, however, which we may for practical pur- 
poses call 5 or 6 times as great as OP — i.e., 5 P.E. or 6 P.E. A 
word spelled by 98 per cent of the pupils becomes more in- 
telligible. It would be located at a point K, a vertical at which 
(KL) would cut off 2 per cent of the area of the entire fre- 
quency surface. The point K will be found to be at a distance 




Fig. 12. Normal Surface of Frequency. 

of about 3 P.E. below the median 0, i.e., at 3 P.E. A word 
spelled by nobody — i.e., a word rated at o — would be at, say, 
+ 6 P.E., and a word spelled correctly by 50 per cent of the 
group would be located at the median 0, that is, at a point 
above and below which are an equal number of cases. 

It will be interesting and will serve to show the misleading 
character of per cent ratings to observe what we mean by 
saying that one word is more difficult than another. Observe 
the two following groups of words taken from Tables III and IV 
for the 3d grade: 

(A) (B) 

Per Cent 
Correct 

tailor 38 beautiful 

lesson 37 beginning 

another 36 telephone 

wear 35 pigeons 



Per Cent 
Correct 

10 

9 



42 Spelling Ability — Its Measurement and Distribution 

According to the ratings of these words the differences in 
point of difficulty between the words of group A are equal to the 
differences in group B, for the differences are all represented 
by i per cent. Habitually we are likely to think that this is true. 
But such a way of thinking quite neglects the form of distribu- 
tion of spelling ability. In fact it assumes that the frequency 
surface is a rectangle — i.e., that there are just as many very 
poor or very good spellers as there are spellers of medium 
ability. This we know is not true. The mediocre are always 
much more numerous than the dull or the gifted. A figure such 
as Fig. 12 takes account of this fact. 

Now the words in group A are much nearer the median 
(which would be a word 50% correct) than are those of group 
B. They are located on the base line at points such that between 
adjacent verticals drawn at these points one per cent of the 
area will lie. The words of group B, more remotely placed 
with reference to the median, are also so situated that between 
their adjacent verticals one per cent of the area will lie. But 
the points for group B stand at greater distances apart than 
do the points for group A because the verticals or ordinates are 
shorter for the remoter group. As a matter of fact, the differ- 
ence in difficulty between " beautiful " and " pigeons " is more 
than twice as much as the difference in difficulty between 
" tailor " and " wear," although each difference is represented by 
the same per cent. 

Bearing in mind the meaning of these per cent values we may 
readily place the 50 words of Tables III and IV along the 
^r-axis or base line of a normal frequency surface. " Even," 
which is rated 59 per cent for the 3d grade, would be at a point 
below the median between whose ordinate and the median ordin- 
ate is 9% of the area of the surface. Calling the median zero and 
referring to Table XIV, we find that 9% of the cases (900 in 
10,000) corresponds to a value of P.E. which lies between .3 
and .35. By interpolation this value is found to be .338. There- 
fore the position of " even " is at — .338 P.E. This may be repre- 
sented on Fig. 13 by the point 1. "Lesson" (37% correct) will 
be at a point above zero between which and zero are 13% of 
the cases of a normal frequency surface. Table XIV locates 
this point at +49 P.E. (Point 2, Fig. 13). " Only " (65% 



Scaling the Words 



43 



correct) is at — .572JP.E. (Point 3, Fig. 13) ; "smoke" (46%) 
at +.148 P.E. (point 4); "pear" (31%) at +.735 P.E. (point 
5) ; " minute " (26%) at +.955 P.E. (point 6) ; " cousin " (19%) 
at + 1.300 P.E. (point 7), and so on. Words rated above 50% 
are located below the median : those under 50% are above the 
median. Their distances from the median are negative in the 
first case and positive in the second. 

Assuming the same form of distribution for the 4th grade 
we find that " even " (79%) is located at — 1.20 P.E., " only " 
(75%) at precisely — 1.00 P.E., and " pear " (42%) at +.30 P.E. 
Similarly for each grade by using the per cents of Table III and 
IV and the P.E. equivalents of Table XIV we may " place " all 




'^2$Z. P£. 3 / % ZStfE.1 +ZPE. 



Fig. 13. Showing the placing of the first 7 words of the Preferred 

List. 3d grade. 



the words. Table XVII gives the per cents and P.E. equivalents 
of the 50 words of the Preferred Lists which from now on will 
be treated as one list. Figs. 14, 15, 16, 17, 18 and 19 show how 
the words appear when arranged on a linear scale for each 
grade. For the meanings of the numbers, each of which refers 
to a word of the Preferred List, see Table XVII or Appendix II. 
Table XVII with its corresponding figures (14 to 19) affords 
standards for grade performances. As will be observed, the P.E. 
values of all the words are calculated for each grade with refer- 
ence to the median of that grade, which is called zero. Their 
use may be illustrated with reference to the 4th grade. We may 
test a pupil of that grade by beginning with the easiest word and 
proceeding to the next hardest and the next and so on. The 
series would run : 1 even, 3 only, 2 lesson, or 5 front, 28 chicken, 
or 41 Tuesday, 4 smoke, 11 pretty, 8 bought. ... By the time 



44 Spelling Ability — Its Measurement and Distribution 



4 



x% 



o 1 



IN 

45 






^ 

^ 



o 


§. 


+ 



o - 

•22 




5$ 


o 




1 


^3 




§ 




O 



:£$ 





"2 


o 


Q '■ 


§ 


oi 


CV 


o 






^Sr 


o 


o 


«><v 


i 


<f 


*s 




1 


-.*• 


o 




S3 


NS 


o 


s£ 





o 

o 1 






r-o o 



5 . 


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o 

CO 
i 


o 


55 


o 


sS - 


*&2 


<D 


-v. 




<M 


1 


*5 


i 


o 




O 


CO 


lj 


<v 


r 


<©::3 


Of 


O 

a ' 


oieo 


o 


o 


- 


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O 




O 


«x*- 




CO 


CVC - 


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fj*<oN 



n«9«rg- 



Scaling the Words 



45 



TABLE XVII 

Per Cents Correct and P.E. Equivalents for Each Word of the 
Preferred List. Grades 3-8. See Figs. 14-19 



No. 

of 

Word 



10 

11 

12 
13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
25 



27 
28 
29 
30 

31 
32 
33 
34 
35 

36 
37 
38 
39 
40 

41 
42 
43 
44 
45 

46 

47 
48 
49 
50 



Words 



even .... 
lesson. . . , 
only .... 
smoke. . . 
front 

sure .... 
pear .... 
bought. . 
another . 
forty 

pretty. . . 
wear, 
button . . 
minute. . 
cousin 

nails. 

janitor . . . 

saucer 

stopping, 
sword. . . . 

freeze. . . . 
touch 
whistle. . . 
carriage . . 
nor 

already. . 
beginning 
chicken . . 
choose . . . 
circus. . . . 

grease. . . . 
pigeons. . . 
quarrel . . . 
saucy 
tailor 

telegram., 
telephone 
tobacco . . 

too 

towel 

Tuesday. . 
tying 
whole. . . . 
against. . . 
answer. . . 

butcher . . 
guess 
instead . . . 
raise 

beautiful . 



3d Year 



P.E. 



— .337 79 

+ .492 72 



4th Year 



— .571 
+ .149 

— .037 

+ .112 

+ .735 

+ .376 

+ .531 

+ .037 

+ .187 
+ .571 
+ .693 
+ .954 
+ 1.302 

+ .261 
+ 1.302 
+ 1.819 
+ .909 
+ 1.670 

+ .820 
+ .187 
+ 1.145 
+ 1.670 

— .492 

+ 1.475 
+ 1.988 
+ .037 
+ 1.145 
+ 1.248 

+ 1.819 
+ 2.188 
+ 1.537 
+ 1.602 
+ .453 

+ 1.537 

+ 2.083 
+ 1.742 
+ 1.602 
+ 1.047 

+ .149 
+ .224 
+ 1.415 
+ 1.302 
+ .909 

+ .652 

+ 1.248 
+ .693 
+ 1.196 
+ 1.900 



P.E. 



-1.196 

- .864 
-1.000 

- .735 

- .864 

- .187 
.299 

- .571 
.261 

- .453 

- .652 
.037 

- .074 
.453 
.112 

• .299 
.299 
.820 
.414 
.149 

.149 

■ .074 

■ .187 
.376 

■ .414 



5th Year 



+ .299 

+ 1.000 

— .778 
+ .612 
+ .414 

+ 1.357 

+ .820 
+ .414 
+ .571 

— .187 



.735 
.571 
.414 
.864 
.224 

.778 
.299 
.261 
.778 
.112 

.337 
.693 
.074 
.149 
.074 



P.E. 



-1.819 
-1.415 
-1.819 
-1 . 537 



6th Year 



— 1 . 248 90 



— .735 

— .299 
—1.196 
— 1 . 145 

— .571 

—1.047 

— .414 

— .414 

— .453 

— .735 

— .820 

— .299 
+ .299 

— .187 

— .261 

— .693 

— .376 

— 224 

iooo 

— .571 

+ .261 
+ .492 
—1.415 

+ .074 
.000 



+ 



.492 
.337 
.112 
.376 
.778 

.414 
.074 
.376 
.909 
.531 

.652 

.778 
.531 
.149 
.652 

.735 
.037 
.453 
.652 

.778 



67 



P.E. 



—2.188 
— 1.« 

—2.439 
—2.305 
—1.900 

—1 . 145 

— .864 
— l.J 

—1 . 602 

— .864 

—1 . 900 

— .954 

— .909 

—1.096 
—1.819 

—1.670 
—1 . 302 

— .299 

— .820 
—1 . 145 

—1.415 
—1.302 

— .531 

— .652 

— .693 

— .453 
+ .149 
—1 . 900 

— .376 

— .864 

+ .571 

— .261 
1.000 

.074 
1.000 

.492 

.652 

—1.000 

+ 1.047 

— .909 

-1 . 248 
- .693 
-1 . 145 
-1.000 



7th Year 



P.E. 



-1 . 602 90 



-1 . 537 
- .652 
-1 . 602 
-1.475 
-1 . 537 



—2.188 
—2.305 
—2 . 789 
—2.597 
—2.305 

—1.819 
—1 . 302 
—2.305 
—2.305 
—1.415 

—1.900 
—1 . 475 

— .954 
—1 . 602 
—1.819 

—2.083 
—1.302 
—1.196 
—1.047 
—1 . 602 

—1.602 
—1.475 
—1.000 
— 1 . 302 
—1.096 

+ .224 

— .612 
—2 . 597 

— .571 
—1.000 

+ .299 

— .778 
— 1 . 602 

— .820 
—1.302 

— .909 
—1.415 
— 1 . 742 
+ .778 
—1.145 

— 1 . 670 
—1.047 
—1.475 
—1.475 
— 1 . 900 

—1.900 

—1.096 
— 1 . 670 
—2 . 188 
—2.305 



8th Year 
% P.E. 



-2.789 
-2 . 597 
-3 . 450 
-3 . 450 
-2.789 

-2.305 
-2 . 305 
-2.789 
-2.597 
—1 . 670 

—2.305 
—2.188 
—1 . 670 
—2.083 
—2.439 

—2.597 
—1 . 900 
—1 . 302 
—1 . 475 
—2.188 

—2.305 
—2.188 
— 1 . 537 
— 1 . 537 
—2.305 

—1.096 
—1.000 
—3 . 450 
—1.357 
—2.439 

— .261 
—1.357- 
—2 . 305 
—1.145 
—1 . 475 

—1.475 
—1.670 
—2.597 
+ .261 
—2.305 

— 1 . 988 
— 1 . 670 

1.900 
—2 . 305 

2.789 

2.789 
—1.537 

1.988 
—2.305 

2.597 



we have reached 13 button, 22 touch, 50 beautiful, 12 wear, and 
48 instead, we are dealing with a group of words which 50 per 
cent of 4th-grade children spell correctly. The performance of 



46 Spelling Ability — Its Measurement and Distribution 

a given 4th-grade pupil should approximate at least the standard 
set by these words. If we are asked, "What is 4th grade spelling 
ability ? " we may answer that it is the ability to spell these 
words that cluster about the median. Of course it is toi be 
expected that any given pupil will miss some of the easier words 
and spell some of the harder words. We should test him by the 
whole series of 50 and his errors for words below the median 
may be balanced against correct spellings of words above the 
median at an approximately equal distance. He may miss 
49 raise ( — .15 P.E.) but spell 20 sword or 21 freeze (+.15 P.E.). 
He may miss 16 nails, but spell 7 pear. In such cases he should 
be credited with having spelled the easier word. 

In a similar way, by using the words in the order in which they 
are placed for any other grade, we may determine whether a 
child is as good a speller as the median children of that grade. 
We do not need, however, to use the median as a standard unless 
we wish to. We may choose + 40 or + 60 and ascertain whether 
children are able to spell up to that point in the same manner as 
is indicated above for the zero-point. It is to be observed, 
however, that our series does not offer a very satisfactory test in 
the higher grades for such standards, because there are so few 
words that are placed as high or higher than + 40 or + 60. 
The words, in short, are not difficult enough for this purpose. In 
a later section of the paper we shall introduce harder words into 
the series precisely with the object of affording a fuller test of 
ability for the higher grades. 

There remains, however, for the present one other use which 
may be made of our data. We may wish to disregard grades 
altogether and seek an answer to the question, In general, how 
hard are these words for children of the elementary school above 
the 2d year? or, with reference to a graphical representation, 
What is the average position of each word on a linear scale — 
that position from which the positions for each grade deviate 
by the smallest amounts? 

To answer such a question we shall have to use one point 
of reference for all grades instead of a different one for each 
grade. In the above treatment we have expressed each word- 
value as a deviation from the median of the particular grade 
we were considering. We shall now use this same data but 



Scaling the Words 47 

transfer the point of reference to the third-grade median by- 
using the median intervals which were derived in Section 1 1. 
In Table XVI (page 39) we have given the results of our inquiry 
into the amounts of these intervals as follows : 



From M 3 to M 4 


1.351 P.E. 


" M 3 " M 5 


2.187 " 


" M 3 « M 6 


3.238 " 


" M 3 " M 7 


3.899 " 


" M 3 " M 8 


4.809 " 



Table XVIII gives, for each of the 50 words, its position for 
each grade when referred to the 3d-grade median as the zero- 
point or point of reference, together with the " average 
position " of each word. The method of securing these figures 
may be illustrated as follows: 

From Table XVII the P.E. values of the word " even " for 
each grade, referred to its own median, are shown to be 



3d grade, 


— .337 


4th " 


—1.196 


5th « 


—1.819 


6th " 


—2.188 


7th " 


—2.188 


8th " 


—2.789 



The first of these values is of course already referred to the 
3d-grade median. To refer the others to the same point we must 
increase each of them by the amount by which each grade 
median stands above the 3d-grade median, i.e., we must find the 
sum (algebraic) of — 1.196 and 1.351, of — 1.819 and 2.187, 
of — 2.188 and 3.238, of — 2.188 and 3,899, and of — 2.789 
and 4.809. These sums give the figures of Table XVIII for the 
word " even." Their arithmetical mean is taken as the average 
position. 

Fig. 20 shows the averages of Table XVIII when reduced 
to a scale. The noticeable thing about these tabular and graphic 
representations is the fact that the words from easiest to hardest 
differ so little. The words "even" and "only" (No. 1 and 
No. 3), which are — .337 and — .571 respectively for the 3d 
grade, appear above the zero point at +.699 and +.569. 
Similarly the word "too" (No. 39) which for the 8th grade 
alone is + 5.07 becomes for all grades only + 3.491. It is a fact 



48 Spelling Ability — Its Measurement and Distribution 

TABLE XVIII 

The Position of Each Word in Each Grade when Referred to the 3d- 

Grade Median as the Zero-point; and the Average Position of 

Each Word for All Grades, when so Referred. 1=P.E. 



Word 
Num- 
ber 

1 
2 

3 

4 
5 
6 

7 

8 

9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

41 
42 

43 
44 
45 
46 

47 
48 
49 
50 



Word 



3d 


4th 


Grade 


Grade 


—.337 


.155 


.492 


.487 


—.571 


.351 


.149 


.616 


— .037 


.487 


.112 


1.164 


.735 


1.650 


.376 


.780 


.531 


1.612 


.037 


.898 


.187 


.699 


.571 


1.388 


.693 


1.277 


.954 


1.804 


1.302 


1.463 


.261 


1.052 


1.302 


1.650 


1.819 


2.171 


.909 


1.765 


1.670 


2.5*00 


.820 


1.500 


.187 


1.277 


1.145 


1.164 


1.670 


1.727 


—.492 


.937 


1.475 


1.650 


1.988 


2.351 


.037 


.573 


1.145 


1.963 


1.248 


1.765 


1.819 


2.708 


2.188 


2.171 


1.537 


1.765 


1.602 


1.922 


.453 


1.164 


1.537 


2.086 


2.083 


1.922 


1.742 


1.765 


1.602 


2.215 


1.047 


1.575 


.149 


.573 


.224 


1.052 


1.415 


1.612 


1.302 


2.129 


.909 


1.463 


.652 


1.014 


1.248 


2.044 


.693 


1.425 


1.196 


1.202 


1.900 


1.277 



5th 
Grade 



6th 


7th 


8th 


Grade 


Grade 


Grade 


1.050 


1.711 


2.020 


1.250 


1.594 


2.212 


.797 


1.110 


1.359 


.933 


1.302 


1.359 


1.338 


1.594 


2.020 


2.093 


2.080 


2.504 


2.374 


2.597 


2.504 


1.250 


1.594 


2.020 


1.636 


1.594 


2.212 


2.374 


2.484 


3.139 


1.338 


1.999 


2.504 


2.284 


2.424 


2.621 


2.329 


2.945 


3.139 


2.142 


2.297 


2.726 


1.419 


2.080 


2.370 


1.568 


1.816 


2.212 


1.936 


2.597 


2.909 


2.939 


2.703 


3.507 


2.418 


2.852 


3.334 


2.093 


2.297 


2.621 


1.823 


2.297 


2.504 


1.936 


2.424 


2.621 


2.707 


2.899 


3.272 


2.586 


2.597 


3.272 


2.545 


2.803 


2.504 


2.785 


4.123 


3.713 


3.387 


3.287 


3.809 


1.338 


1.302 


1.359 


2.862 


3.328 


3.452 


2.374 


2.899 


2.370 


3.809 


4.198 


4.548 


2.977 


3.121 


3.452 


2.238 


2.297 


2.504 


3.164 


3.079 


3.664 


2.238 


2.597 


3.334 


2.746 


2.990 


3.334 


2.586 


2.485 


3.139 


2.238 


2.157 


2.212 


4.285 


4.677 


5.070 


2.329 


2.754 


2.504 


1.990 


2.229 


2.821 


2.545 


2.852 


3.139 


2.093 


2.424 


2.909 


2.238 


2.424 


2.504 


1.636 


1.999 


2.020 


1.701 


1.999 


2.020 


2.586 


2.803 


3.272 


1.636 


2.229 


2.821 


1.763 


1.711 


2.504 


1.701 


1.594 


2.212 



Average 
Position 



even 

lesson. . . . 

only 

smoke 

front 

sure 

pear 

bought. . . 
another . . 
forty 

pretty 

wear 

button . . . 
minute. . . 

cousin 

nails 

janitor . . . 

saucer 

stopping. . 
sword. . . . 

freeze 
touch 
whistle. . . 
carriage . . 

nor 

already. . . 
beginning 
chicken . . 
choose . . . 
circus 

grease. . . . 
pigeons. . . 
quarrel. . . 
saucy 
tailor 
telegram . 
telephone 
tobacco . . 

too 

towel 

Tuesday. , 

tying 

whole 
against. . . 
answer. . . 
butcher . . 
guess 
instead. . . 

raise 

beautiful . 



.368 

.772 

.368 

.650 

.937 

1.452 

1.888 

.991 

1.042 

1.616 

1.140 
1.773 
1.733 
1.734 
1.452 
1.367 
1.888 
2.486 
2.000 
1.926 

1.494 
1.811 
1.963 
2.187 
1.616 
2.448 
2.679 
.772 
2.261 
2.187 

2.679 
2.524 
2.075 
2.563 
1.409 
2.601 
2.261 
1.811 
3.096 
1.656 

1.535 
1 

1 
2 

1 
1 
2 
1 

1 



409 
656 
038 
535 
452 
224 
734 
535 
409 



Scaling the Words 



49 



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50 Spelling Ability — Its Measurement and Distribution 

then that for these words the influence of higher grades is to 
make easy words harder and of lower grades to make hard 
words easier. That is, grade considered, these words are harder 
for children of the upper grades than they are for those of 
the lower grades. There are at least two reasons for this 
condition. 

First, as a rule these particular words are taught in the 
lower grades. A popular speller, taken at random, presents 
31 of the 50 words in the 3d year's work, 10 in the 4th, 2 in 
the 5th, and none in higher grades. There is nothing to lead 
one to suppose that this is peculiar. The words were among 
those chosen, it will be remembered, as at least in the speaking 
vocabulary of 3d-grade children. Most of them if taught at all 
Will be taught in that grade. We may assume therefore that 
the 3d-grade record is somewhat affected by the recency with 
which these words have been presented. The succeeding grades 
will to some extent be discriminated against in the record. 

Second, the necessary basis of selection for these words from 
the larger lists would make it impossible for the words to take 
the same position on the scale for all grades. Consider a word 
which was spelled correctly by 50% of the 3d-grade children. 
Such a word would be at M s . In order to take the same position 
on the 8th-grade record it must be as far below M 8 as is the 
distance, already determined, between M s and M 8 or — 4.809 
P.E. To do this it would have to be spelled correctly by 9994 
pupils out of 10,000 (Table XIV), i.e., it would be 100% correct. 
But such a word would not have been selected, because it is not 
difficult enough in the 8th grade to be of any value as a test of 
ability. On the other hand, a word missed often enough in the 
8th grade to be satisfactory as a test (say, 90% correct) would 
have to be less than 1% correct on the 3d-grade record in order 
to take the same position on the scale. Such a word would have 
been of no use as testing 3d-grade ability and would have been 
rejected. 

The fact is that the span from 3d to 8th grade is — if our median 
distances be correct — too great for any list of words to be in all 
respects satisfactory. We need several lists each of which shall 
be given to three or four consecutive grades and overlapping 
on one another — e.g., one for 2d, 3d, and 4th grades, another 



The Use of the Scale 



Si 



for 3d, 4th, 5th and 6th grades, and another for 5th, 6th, 7th 
and 8th grades. An attempt will be made in a later section to 
do this and to show the results that may be expected. 

§ 13. The Use of the Scale 
Meanwhile, however, we venture to think that the scale as 
,shown in Fig. 20 is important and valid within its range. It 
may be used in several ways of which at least three are 
important. 

1. It may be used just as it is without reference to the fact 
that the words are not separated from each other by equal 
intervals. We know the value or weight to assign to each word. 
We shall therefore not make the mistake of assuming that all 
the words are of the same value, as is the usual school practice. 

2. Certain words of the series may be used which differ from 
each other by approximately equal steps. 

TABLE XIX 

Words Arranged in Order of Difficulty According to the Scale 

and Their P.E. Values 



No. on 
Scale 


Word 


P.E. x 100 


No. on 
Scale 


Word 


P.E. x 100 


3 


only 


57 
70 
84 
90 
106 
108 
114 
117 
131 
138 
147 
155 
157 
159 
165 
165 
168 
168 
171 
174 
176 
176 
184 
187 
187 


14 
7 
40 
38 
43 
13 
17 
33 
44 
30 
20 
23 
19 
24 
47 
37 
29 
36 
18 
34 
26 
32 
27 
31 
39 


minute 

pear 


194 


1 


even 


196 


4 


smoke 


towel 


198 


28 
5 


chicken 

front 


tobacco 

whole 


199 
202 


9 

2 


another 

lesson 


button 

janitor 

quarrel 

against 

circus 


203 
205 


8 
11 


bought 

pretty 


207 
211 


16 


nails 


214 


46 


butcher 

Tuesday 

sure 


sword 


219 


41 
6 


whistle 

stopping 

carriage 

guess 


219 
221 


45 
25 


answer 

nor 


234 
236 


49 


raise 


telephone 

choose 

telegram 

saucer 


241 


15 


cousin 


250 


50 
22 


beautiful 

touch 


255 

260 


21 


freeze 


saucy 


267 


10 


forty 


already 

pigeons 

beginning 

grease 


270 


48 
12 


instead 

wear 


274 
292 


35 


tailor 


329 


42 


tving 


too 


349 











Spelling Ability — Its Measurement and Distribution 



3. Small groups of words may be so selected as to be 
equally difficult as groups ; or they may be so selected that their 
group-difficulties constitute an ascending series from easy to 
hard, differing by equal amounts. 

1. By the first of these methods the entire series would be 
utilized or so much of it as in any given case would thoroughly 
test the subject. The order of the words of the series as given 
in Figure 20 is shown in Table XIX in the first column and in 
the second column the test values or weights of these words 
are given. 

2. If it is desired to use a scale whose words differ in 
difficulty by equal steps, the arrangement as shown in Table 
XX will be found convenient. 

TABLE XX 
A Ten-point Scale 



No. of 

Word 

(Fig. 20) 



Word 



P.E. x 100 



3 

4 

9 

11 

45 
35 
30 
37 
34 
27 



only 

smoke 

another . . 

pretty 

answer. . . 
tailor 
circus .... 
telephone, 
saucy .... 
beginning. 



57 
84 
108 
131 
159 
187 
214 
241 
267 
292 



27 
24 
23 
28 
28 
27 
27 
26 
25 



To this series may be added 39 " too " whose P.E. x 100 is 
349 and which differs from " beginning " by 57 or approximately 
two steps. 

In the series of Table XX the average step is 26.2 with an 
A.D. of 1.3; or if the word " too ■■" is included the average step 
is 26.6 with an A.D. of 14. This is quite accurate enough for 
any use to which the scale is likely to be put. If this conclusion 



The Use of the Scale 53 

is accepted, these eleven words may be used to express our 
judgments of other words concretely and in terms that every- 
body can understand. We should not then have to resort to 
such terms as " hard," " easy," " rather difficult," " very hard," 
etc., but we may judge a word to be " as hard as ' another,' " 
" equal in difficulty to ' beginning,' " " as hard as ' answer ' but 
not as hard as ' tailor,' " etc. It is very desirable that other 
words should at some time be added to the scale at both ends. 
There are many words harder than " beginning " or " too " and 
there are others easier than " only," although the latter do not 
constitute much of a school problem. Neither set, however, 
could be used over a range as wide as 3d to 8th grades. 

3 (a). It is often desirable to offer tests of equal difficulty, 
but of different words at various intervals of time to the same 
group or to the same individual. We may thus secure a progress 
record. In spelling, however, this has proved to be very difficult 
if not impossible. We can never be sure that the second or 
third test is equal in difficulty to the first test. In fact we may 
be pretty sure it is not. To give the same words over again is 
often valueless because of the added special familiarity with 
them. The following lists therefore are offered as lists of equal 
difficulty. The sum of the P.E. values in each is 976 or 977. In 
using them the words may be weighted as indicated, or may, with 
no great loss in precision, be each given a credit of 1. 

Number Number 

in Preferred in Preferred 

List Weight List Weight 

Group A Group B 

41 Tuesday 16 45 answer 16 

10 forty 18 48 instead 18 

40 towel 20 43 whole 21 

44 against 22 17 janitor 21 

47 guess 24 24 carriage 24 

Group C Group D 

49 raise 17 21 freeze 18 

22 touch 17 12 wear 19 

42 tying 19 7 pear 20 

14 minute 20 13 button 21 

18 saucer 27 20 sword 22 

Group E Group F 

16 nails 14 8 bought 12 

46 butcher 15 11 pretty 13 

15 cousin 17 19 stopping 23 

29 choose 26 37 telephone 25 

32 pigeons 28 34 saucy 27 



54 Spelling Ability — Its Measurement and Distribution 



(b) It may also be desirable to test, not with single words, 
which only in the long run may be expected to conform to the 
positions assigned to them, but with groups of words whose 
difficulties as groups differ by constant amounts. Such a series 
of groups arranged from easy to hard would themselves con- 
stitute a scale — a sort of Binet-Simon scale for measuring ability 
in spelling. On the analogy of the Binet-Simon scale we might 
easily fix upon a certain minimum performance for a group at 
which or better than which a subject might be allowed to have 
" cleared " that group and might pass on to the next. He might 
also be given additional credits for spelling words in groups 
above the highest one which he cleared. 

The groups are arranged in-order of difficulty, Group I being 
the easiest and Group VII the hardest. Within each group the 
four words are also arranged in their order of difficulty begin- 
ning with the easiest. Since, however, within each group the 
words differ little in difficulty, they may be taken as having 
equal weights without material error. It is true that Group VII 
is not nearly as satisfactory in this respect as the others, differing 
between the first and fourth words by 1.08 P.E., whereas the first 
six groups have a range of but .225 on the average. 

Group I P.E. xlOO Group IV P.E. x 100 

3 only 57 10 forty 176 

1 even 70 12 wear. 

4 smoke 84 42 tying 



28 chicken 



90 



38 tobacco 



184 

187 
199 



Average. 
Group II 

5 front 

9 another 

2 lesson 

8 bought 



75 




P.E. x 100 




106 


33 


108 


30 


114 


24 


117 


47 



Average. 
Group V 

quarrel 

circus 

carriage 

guess 



Average. 
Group III 

16 nails 

46 butcher 

41 Tuesday 
6 sure 



Ill 
P.E. x 100 
138 
147 
155 
157 



Average. 
Group VI 

29 choose 

36 telegram 

34 saucy 

26 already 



186.5 
P.E. x 100 
207 
214 
234 
236 

223 
P.E. x 100 

250 
255 
267 
270 



Average . 



149 
Group VII 
37 telephone . 
32 pigeons . . . 
31 grease 
39 too 



Average. 
P.E. x 100 
241 

274 
329 

349 



260.5 



Average . 



298 



The Zero Point of Spelling Ability 55 

The average differences in difficulty between these groups in 
succession are 36, 38, 37.5, 36.5, 37.5 and 37.5. This is probably 
the most important use of the scale, for present school practice. 

If it is true that the general scale (Table XVIII and Fig. 20) 
may be used in these three ways — as a whole, by words selected 
to be at equal intervals, and by grouping words so that the groups 
are equal or differ by equal amounts — then it is also true that 
each of the grade scales (Figs. 14-19) may be used in like 
manner each for the grade to which it applies. It is probably 
true, moreover, that the grade scales will more closely fit real 
conditions in any given instance than will the scale for all grades. 
The labor of making selections and groupings of words for 
these scales is not great and may be made by any one on the 
analogy of the method used above. 

§ 14. The Zero-Point of Spelling Ability 

As has been suggested in previous sections, we have only suc- 
ceeded in scaling by means of these 50 words a limited segment 
of the entire projection representing spelling ability. Our list is 
essentially an easy list, testing that ability only to a moderate 
degree. Words like " fatiguing," " guarantee," and " conscien- 
tious " (Rice Sentence Test) would stand much higher in the 
scale and require a considerable extension of it to the right; 
while such unfamiliar words as " eurycerous," " delitescence," 
and "gallinaceous" (Klein, ? I2, pp, 388, 389) would take still' 
higher positions, passing quite beyond the range of the ability of 
elementary-school children. 

On the other hand, our scale is as certainly limited at the low 
end. There are many easier words than any we have used so far. 
Such words would reach far down on the scale towards the place 
where the absolute zero-point lies. But they would have been 
totally unfit for use in the higher grades. In fact, with the wide 
range of ability between 3d and 8th grades, it is surprising that 
we find any words at all which will afford a test at both 
extremes. 

Without seeking to determine the limit of the high end of 
the scale — perfect spelling ability — it is quite possible, and 
theoretically very desirable, to find the limit of the low end, i.e., 



56 Spelling Ability — Its Measurement and Distribution 

to find the point where spelling ability just begins to be a positive 
quantity. 

How far, then, below the 3d-grade median, which has hitherto 
been our point of reference, is the absolute zero-point? 

In order to answer this question, a test was given to children 
of the 2d, 3d, and 4th grades. It consisted of 50 words in 
sentences. Nineteen of these had already been used in the 
Selected List (100 word list) ; and, of these, 6 had been chosen 
for the Preferred List. They had all been spelled, about 40 per 
cent or more correct, by the third-grade children. The remain- 
ing 32 words were thought to be among the easiest in the 
language: he, is, on, the, to, of, for, day, etc. 

They were put into sentences as follows and dictated at 
schools II and VIII: 



Easy 50-W0RD Test 

You will hear him coming. 

He is on the road and is almost sure to pass in front of me. 

I send for him every day. 

Go into the school. 

But do not touch the table. 

He also has only one pair of shoes. 

They are not at all pretty. 

No man ought to steal even a penny. 



It seems clear that a child who cannot spell any one of these 
words has substantially no spelling ability. Since our study is 
limited to written words we shall say, therefore, that for our 
purpose a child who does not, save by chance, write a single 
word of this list so that it can be recognized as correctly spelled 
has no ability. 

On account of the marked improvement in spelling of children 
in the latter half of the second school year over those in the 
first half of that year, we have treated the two half-years of 
the 2d grade separately, calling the lower half 2a and the upper 
2b. We shall proceed as follows. We shall find the distance 
between the 3d-grade median and the 2&-grade median and the 
distance between the latter and the 2a-grade median. Then if 
there are children of the 2a grade who utterly break down and 



The Zero Point of Spelling Ability 



57 



fail to write any word correctly we shall find their place in the 
2a distribution. 

Table XXI shows the records of individual pupils according 
to their rating in the Easy 50- Word Test. Note the fact that 
no children of the 4th, 3d, or 2b grades wholly failed in this 
test. In 2a, however, 39 children were rated 10% or less, and 
of these there were 8 pupils who were actually marked zero. 
This is 4.6% of all the children of 2a. 



TABLE XXI 
Distribution of Individual Ratings. Easy 50-word Test 
Table reads: in 2a 39 children, or 22%, were rated between and 10%; 
32 children, or 18%, were rated between 11% and 20%, etc. In 26 5 chil- 
dren, or 3%, were rated between 11% and 20%, etc. 



Per Cent Correct 


2a Grade 


26 Grade 


3d Grade 


4th Grade 


No. 


% 


No. 


% 


No. 


% 


No. 


% 


0- 10 


39 
32 

37 

27 

18 

14 

5 

2 

1 




22 
18 
21 
16 
10 
18 

3 

1 
.6 





5 

9 
29 
26 

47 
31 
14 

7 
1 



3 
5 

17 
15 

28 
18 

8 

4 
.6 




1 

4 
7 
11 
25 
33 
36 
30 
21 



.6 

2 

4 

7 
15 
20 
21 
18 
13 










4 

13 

29 

50 

86 

134 





11-20 





21- 30 





31- 40 





41- 50 


1 


51- 60 


4 


61- 70 


9 


71- 80 


16 


81- 90 


27 


91-100 


42 






Totals 


175 




169 




168 




316 




Medians 




26.50 




56.17 




72.50 




88.12 



The medians for the grades are as follows : for 2a, 26.50% ; 
for 2b, 56.17%; for 3d grade, 72.50%; and for 4th grade, 
88.12%. The rapid rise of spelling ability from low second 
through the fourth grade is remarkable. It is much greater 
than the improvement during the next four years. Although 
the interval in time between 2a and 2b is but half a year, the 
medians suggest that the increase in ability between these grades 
is greater than it is between any consecutive yearly grades above 
the second. Further analysis will more precisely confirm this 
inference. 

Proceeding as in the case of grades 3 to 8, we show in 
Table XXII the amount and per cent of overlapping of each 



58 Spelling Ability — Its Measurement and Distribution 



grade beyond the medians of the other grades, together with 
the corresponding linear segment in terms of the Probable Error 
as a unit. 

TABLE XXII 

Amount and Per Cent of Overlapping with P.E. Equivalents. 
Easy 50-word Test 



4th Grade 

o 
o 

? 

4 

2.37 

2.9395 

30 

17.86 
1.3649 



2a grade. 

m 

26 grade. 
3d grade. 
4th grade 





2a Grade 


2b Grade 


3d Grade 


No. 

% 
P.E. 




17 

9.71 

1.9254 


3 

1.71 

3.1429 


No. 

% 
P.E. 


160 
94.67 
2.3932 




22 

13.02 
1.6690 


No. 

% 
P.E. 


165 
98.21 
3.1143 


140 
83.33 
1 . 4341 




No. 

% 
P.E. 


316 

100 

? 


308 
97.47 
2.8976 


262 
82.91 
1.4094 



From these results, Table XXIII is computed. The object in 
this table is to show how, by using all the data of Table XXII, 
various values of the median intervals may be obtained whose 
averages will be the most probably correct values. The interval 
between the medians of 2a and 2b is written M.2a-2b ; that 
between the medians of 2b and the 3d grade is written M2&-3 ; etc. 

It may be remarked parenthetically that in the number 1.3771 
of Table XXIII for the difference between the 3d- and 4th- 
grade medians, we have a striking confirmation of the substan- 
tial accuracy of our results as shown in Table XVI. The 
corresponding number is there given as 1.3505. That these should 
differ by so little when carried out upon different test material 
is exceedingly satisfactory. 

According to Table XXIII, the 2&-grade median is approxi- 
mately 1.35 P.E. below the 3d-grade median. The 2a-grade 
median is about 1.87 P.E. further below, or 3.22 P.E. below the 
3d-grade median which we have thus far used as our origin or 
point of reference. 

But we have not yet reached the point of zero ability. Typical 



The Zero Point of Spelling Ability 



59 



TABLE XXIII 



Values of Median Intervals and Their Derivation 
(2o-4th Grade) 


^2a-2& 


^26-3 


^3-4 


1 . 9254 
(direct) 




1.6690 
(direct) 


1.3649 
(direct) 


2.3932 

(direct) 




1.4341 

(direct) 


1.4094 
(direct) 


1.4739 
(M 2a _ 3 — M 2b _ 3 ) 




1 2175 

(^2a- 3 — ^2a-2&) 


1.2705 
(^26-4—^26-3) 


1 . 6802 

(^3a-2— M 3-2b) 




1.5746 
(M 26 _ 4 -ikf 3 _ 4 ) 

.7211 

(^3-2a— ^26-2a) 

1.4882 
(M 4 _ 2b —M 4 _ 3 ) 


1.4635 

(^4-26—^3-26) 


Averages 1 . 8682 




1.3518 


1.3771 



2a children have some ability, namely, according to our record, 
an ability to score 26.5% in the Easy 50-Word Test. The 
children of that grade who were unable to write any word 
correctly were 8 in number, representing 4.6 per cent. These 8 
are included in the 39 rated between zero and 10% (Table 
XXI). Assuming that 2a children are grouped about their 
median according to the " normal " distribution, we find that 
in order to cut off 4.6% from the low end we must take a point 
2.5 P.E. below the median, (See Table XIV). This brings the 
zero-point at 5.72 P.E. below the 3d-grade median (3.22 + 2.5). 
This figure, 5.72 P.E., can only be taken as approximately 
correct. It would be somewhat influenced by an increase of the 
number of children tested. There is, however, no reason to 
suppose that the children ir schools II and VIII were unusual. 
The testing in grades 3 to 8 in all other schools shows that 
results in these two schools do not materially differ from the 
general results. In both central tendencies and variabilities they 
are a fair average. There seems, then, to be no good reason 
why we should not consider the ratings of children in these 



60 Spelling Ability — Its Measurement and Distribution 

schools as typical. It must be borne in mind, however, that the 
classification of children into grades is a broad one. Just as we 
found it necessary to treat 2d-year children in half-yearly 
sections, so we should find that testing at the beginning even 
of a 20-week term would yield results, especially in the low 
grades, quite different from those obtained by testing toward 
the close of the term. Accordingly, the middle of the term is 
the best time at which to find typical conditions. Moreover, 
in order that the results may be comparable, the testing of all 
grades should be done at the same time. If 2a children were 
tested towards the end of their term in that grade, while 2& 
children were tested towards the beginning of theirs, the median 
interval would be unduly shortened. A considerable addition 
to the reliability of our results is found in the fact that all 
children were tested during the ioth week of a 20-week term. 

We may therefore conclude that the intervals between grades 
2a, 2b } 3 and 4 are substantially as found in Table XXIII. But 
as to the distance of the zero-point below the 2<z-grade median, 
we cannot be precise. Four and six-tenths per cent of the 
2a children got no word right. As many as 22 per cent wrote 
less than 6 words correctly. Some of them probably spelled 
these few simple words correctly by mere chance. If this were 
true, they would have practically no spelling ability. The 
situation is more complicated than the above analysis indicates. 
If we were to assume that all the children who wrote 0-5 words 
correctly had practically no spelling ability (written), the zero- 
point would then be but 1.15 P.E. below the median instead 
of 2.5 P.E. If we were to assume that some of these children — 
say those who wrote no more than 3 words correctly — had zero 
ability, we should find that 29 of the 39 in Table XXI were 
included. Such an assumption would place our zero-point at 
1.44 below the median. There are reasons for thinking that 
this is not far from the true position. The best judgment, there- 
fore, that we can make from our data and from our knowledge 
and experience of school conditions is, that the zero-point is 
about 1.5 P.E. below the 2a median, or about 4.72 P.E. below 
the 3d-grade median. 

We may summarize our results, then, in Table XXIV and 
Fig. 21, as follows: 



Observations on the Distributions Shown in Fig. 21 



61 



TABLE XXIV 

Median Intervals. Zero to 8th Grade Median 





Successive 
Intervals 


Distance 
Above 


2a grade 


1.50 

1.87 

1.35 

1.35 

.84 

1.05 

.66 

.91 


1.50 


26 " 


3.37 


3d " 


4.72 


4th " 


6.07 


5th " 


6.91 


6th " 


7.96 


7th " 


8.62 


8th " 


9.53 







Fig. 21, page 62, shows these facts graphically. 

§ 15. Observations on the Distributions Shown in Fig. 21 

It is to be remembered that in Fig. 21 the eight surfaces of 
frequency constructed on each median vertical are theoretical 
and not according to the record. Moreover, they express the 
assumptions that for each grade the distribution of ability in 
spelling is strictly " normal " and that the real variability is 
alike in all grades. In a later section we shall take up the matter 
of applying to our results distributions which are not normal. 
Meanwhile, however, it will be interesting to observe how satis- 
factory a strictly normal form of distribution proves to be. To 
the extent that it expresses the same or nearly the same facts 
as the record (so far as it should, if valid, do so), it shows its 
value. 

1. In Fig. 21 the 2a surface of frequency does not reach the 
4th-grade median ; but it only falls short a little. According to 
the record in the Easy 50- Word Test no 2a child did as well 
as the median 4th grade child. But the best 2a record was 82% 
which is only a little less than M 4 (88.12 by Table XXI). 

2. In the graphic showing the 3d-grade distribution does not 
quite reach the 8th-grade median. Similarly the record shows 
that no 3d-grade child obtained a score equal to 94.68 which 
(Table XI, p. 2j) is M 8 for the Selected 100- Word List; al- 
though 3 third-grade pupils had scores in the 91 to 95 group. 
(Table XI, column 2.) 

3. By the figure we see that the low end of the 8th-grade 
distribution falls short of M 3 but not of M 4 . In the record the 



62 Spelling Ability — Its Measurement and Distribution 



I 






O' 



■C 11 w 



■H w. rt 53 
c 32 « 

en C .~ _ 

5 « B 

Jjn « as 

6 - 

■STls 



2 « 



-i.8 ^*- 



^^ 



bfl w <u t o « £ 

"g^Illll! 



c3 a o-^ «j n^j 

o^ o^ £?£ g 



bfl 



en >>•*-> <i> m || 

2^ flj"- 



3 M 



u o 



H IT) N 



.353*0.5 fig 

P< en 






Observations on the Distributions Shown in Fig. 21 63 

same is true, although the poorest 8th-graders surpassed the 
median of the 3d grade by more than the figure would indicate 
(See Table XI, page 27). 

4. The 2b curve at its low end does not reach zero (record). 
No 3d-grade child was rated o in the Easy 50- Word Test (Table 
XXI, p. 57). 

5. The most remarkable thing in the figure is the fact that 
the high end of the 2&-grade distribution extends above the low 
end of the 8th-grade distribution. Even with all the recent 
information about retardation, acceleration, mental defect, and 
precocity combined with mis-grading and forced promotions, 
some critics will be hardly prepared to believe that any children 
classified in the first half of the second year of school life can 
do as well in spelling as the poorest who are classified in the 
8th (last elementary) year. This, however, the figure shows; 
and there is good evidence in our record to support it. We 
cannot compare these extreme grades directly because they did 
not write the same test. But in Table XI (page 2j) we observe 
that for the Selected List one 8th-grade pupil is in the 56-60 
group. As a matter of fact his paper was rated at exactly 56%. 
Therefore all the 3d-grade children in groups above 51 to 55 did 
as well or better. This proves to be 22.5% of the 3d-grade 
children. It is remarkable that between one-quarter and one- 
fifth of our 3d-grade children do as well or better than the 
poorest 8th-grade child. But this is not all. In the Easy 50- 
Word Test the highest 2a score was 82%. Only 38 out of 168 
3d-grade children, or 22.6%, did better. See Table XXI, page 
57. (The one child in group 81-90 in the 2a column was 
rated 82. In the 3d-grade column, of the 30 in group 81-90 and 
the 21 in group 91-100 combined, 38 were rated above 82%.) 
We therefore find that the best 2a child and the poorest 8th- 
grade child are equalled or surpassed by the same group of 
3d-grade children, — i.e., 22^4%. Hence the best 2a child and 
the poorest 8th-grader of our record do have the same ability. 
To show this fact, the 2a curve should pass slightly beyond the 
low end of the 8th-grade curve, as in fact it does. This may 
seem to be drawing over-fine conclusions, but it is probable that 
the real overlapping is as great as the record shows. If, out 
of so few 2a children (175), one was found who scored 82%, 



64 Spelling Ability — Its Measurement and Distribution 

it is likely that, with a much larger number, there would not only 
be more children at 82% but some at even higher rating. 
Similarly it is likely that if a much greater number of 8th-year 
children had been tested some would have obtained less than 
56%. To the extent that either one or both of these proba- 
bilities were true, the overlapping would somewhat exceed that 
which the record suggests. 

6. After the argument of the last paragraph it need hardly 
be said that there is both in the graphic representation by theory 
and in the actual record an overlapping of every grade distribu- 
tion on every other from low 2d to 8th. 

The location of the zero-point enables us to draw some inter- 
esting conclusions which were not possible before. A few of 
these will be briefly stated. Taking the medians of each grade 
as indicating typical abilities, regarding the estimated zero-point 
as the true one, assuming normal distributions and equal real 
variabilities for all grades, and defining " to spell twice as well " 
as "to spell words of twice as much difficulty/' Fig. 21 shows 
that 2b children are more than twice as good spellers as 2a 
children, and that 3d-graders are about three times as capable, 
and 4th-graders 4 times as capable. Fifth-grade children spell 
twice as well as 2b children. Eighth-grade children are only 
twice as good as 3d-grade children. This last statement means 
that typical children who have reached the 3d grade have half 
as much spelling ability as is required of the average child in the 
last year of the elementary school. 

We have in Fig. 21 a representation by which the entire range 
of difficulty of spelling words, appropriate to the elementary 
school, may be shown. The notation below the base-line shows 
the positions within that range taken by the 10 words of our 
scale which stand at equal intervals upon it (Table XX, 
page 52). Since these words include both the easiest word 
(" only ") and the hardest (" too ") of the entire scale they show 
its total spread. It will therefore be seen to what extent the 
statement is true which was made at the beginning of Section 
14 to the effect that with these 50 words we have only suc- 
ceeded in scaling " a limited segment of the entire projection 
representing spelling ability." By actual measurement it appears 
that this segment is but a trifle more than one-fifth of the entire 



Supplementary Testing at Schools VI and VII 65 

projection. It will now, however, become apparent that no 
greater segment could have been so scaled reliably from the 
limited material at our disposal. By reference to Fig. 21 it will 
be seen that no words easier than " only " could have been used 
and still have been clearly within the 8th-grade distribution. On 
the other hand no word that scales much higher than " too " 
could have been used and still have been clearly within the 3d- 
grade distribution. With, say, ten thousand children of each 
grade tested each with a longer list, a wider spread could have 
been obtained. The series of fifty words which we used spreads 
over nearly the whole of the base-line common to both the 3d- 
and 8th-grade curves. 

It is evident, therefore, that much remains to be done to perfect 
a scale which shall pretend to completeness. Some of this fur- 
ther scaling will be undertaken in a later section of this mono- 
graph. A great deal must be left for later studies. A great 
many more words must be used both to fill in the gaps within 
the present scale and to extend its limits. Our main purpose has 
been to show the theory and technique required. 

§ 16. Supplementary Testing at Schools VI and VII 
After the data thus far given were in hand the same test 
material was used in two other schools. The results of this 
supplementary testing are now given. The Selected List (100 
words) was dictated during the fall term of 1912 at schools 
VI and VII to 1770 children. Two of the assistants in psy- 
chology at Teachers College acted as examiners, and the papers 
were then scored for individual ratings, but not for word ratings. 
Table XXV gives the distribution of these ratings and the grade 
medians for these two schools. Table XXVI gives the com- 
parisons by grades of the combined results in schools VI and VII 
with those in schools II, III, IV, and V taken together. The 
comparison shows that in general schools VI and VII did not do 
so well as those tested earlier. On the average the grade medians 
are nearly 6y 2 per cent lower. 

To one who would expect close conformity to our previous 
individual ratings in the case of any school taken at random, 
this discrepancy will be disappointing. But to one who recog- 
nizes the wide variability among schools in every subject, the 



66 Spelling Ability — Its Measurement and Distribution 



TABLE XXV 

Distribution of Individual Ratings, Schools VI and VII. 
Selected (100) List 



Per Cent 
Correct 


3d Grade 


4th Grade 


5th Grade 


6th Grade 


7th Grade 


8th Grade 


No. 


% 


No. 


% 


No. 


% 


No. 


% 


No. 


% 


No. 


% 


0- 10 


69 


22.7 


10 


3.4 


1 


.3 


3 


1.2 














11- 20 


58 


19.1 


32 


10.8 


5 


1.7 


5 


2.0 


2 


.6 








21- 30 


65 


21.4 


29 


9.8 


12 


4.1 


8 


3.2 


2 


.6 








31- 40 


49 


16.1 


34 


11.5 


24 


8.3 


7 


2.8 


5 


1.5 


2 


.7 


41- 50 


29 


9.5 


40 


13.5 


31 


10.6 


13 


5.2 3 


.9 


1 


.3 


51- 60 


17 


5.6 


51 


17.6 


41 


14.1 


23 


9.2 


13 


3.9 


3 


1.0 


61- 70 


10 


3.3 


39 


13.2 


47 


16.2 


28 


11.2 


41 


12.2 


15 


5.1 


71- 80 


6 


2.0 


29 


9.8 


53 


18.3 


51 


20.4 


56 


16.6 


16 


5.5 


81- 90 


1 


.3 


25 


8.4 


58 


20.0 


51 


20.4 


114 


33.8 


97 


33.1 


91-100 








7 


2.4 


18 


6.2 


61 


24.4 


101 


30.0 


159 


54.3 


Totals... 


304 




296 




290 




250 




337 




293 


M ■ 


Medians. 




25.27 




51.75 




67.67 




79.36 




84.86 




91.96 



TABLE XXVI 

Comparison of Results Obtained in Schools VI and VII with 
Those Obtained in Schools II, III, IV and V 
Figures show per cent of pupils in each grade who were rated 0-10, 11-20, 



etc. 





3d Grade 


4th Grade 


5th Grade 


6th Grade 


7th Grade 


8th Grade 


Per Cent 
Correct 


Sch. 
II, 
III, 

IV, V 


Sch. 

VT, 
VII 


Sch. 
II, 
HI, 

IV, V 


Sch. 

VI, 
VII 


Sch. 
II, 
III, 

IV, V 


Sch. 

VI, 
VII 


Sch. 
II, 
HI, 

IV, V 


Sch. 

VI, 
VII 


Sch. 
II, 
III, 

IV, V 


Sch. 

VI, 
VII 


Sch. 
II, 
III, 

IV, V 


Sch. 

VI, 
VII 


0- 10 

11- 20 

21- 30 

31-40 

41- 50 . . 

51- 60 

61-70 

71-80 

81- 90 

91-100 


6.9 

15.2 

20.5 

16.1 

13.0 

11.2 

9.6 

4.7 

1.8 

.7 


22.7 

19.1 

21.4 

16.1 

9.5 

5.6 

3.3 

2.0 

.3 




.4 

4.7 

7.7 

12.0 

13.5 

12.4 

14.6 

17.1 

12.7 

5.0 


3.4 

10.8 

9.8 

11.5 

13.5 

17.6 

13.2 

9.8 

8.4 

2.4 


.6 

.6 

3.5 

4.6 
8.9 
10.1 
17.8 
21.0 
19.6 
13.2 


.3 

1.7 

4.1 

8.3 

10.6 

14.1 

16.2 

18.3 

20.0 

6.2 






.5 

.5 

2.4 

5.0 

8.4 

19.6 

30.6 

33.1 


1.2 

2.0 

3.2 

2.8 

5.2 

9.2 

11.2 

20.4 

20.4 

24.4 








.5 

.8 

2.2 

3.8 

13.4 

32.0 

47.1 




.6 

.6 
1.5 

.9 
3.9 
12.2 
16.6 
33.8 
30.0 












.4 

1.5 

6.9 

21.7 

69.7 








.7 

.3 

1.0 

5.1 

5.5 

33.1 

54.3 


Medians . . . 


35.80 


25.27 


60.70 


51.75 


73.10 


67.67 


84.90 


79.36 


90.50 


84.86 


94.68 


91.96 



difference will occasion no surprise. We should do well also to 
bear in mind not only that schools do vary greatly, but that in 
this particular instance there was a constant factor tending to 
lower the ratings. In the supplementary test the dictation was 
given by a stranger; in the original test by the class teacher, 
guided by printed directions. Leaving out of account any sugges- 



Supplementary Testing at Schools VI and VII 67 

tion of unfair methods on the part of teachers for the purpose of 
making a showing, this fact is quite sufficient to account for a 
falling off of results in schools VI and VII without supposing 
them to be much, if any, inferior to the others in the ability of 
their children to spell. The teacher has but one class to examine 
and she takes her time. She doubtless takes full advantage of 
the direction permitting the reading of a sentence " in whole 
or in part as many times as may be necessary to secure its full 
comprehension." She knows her class. The peculiarities or 
defects of pupils are her daily concern and she modifies her 
appeal accordingly. The children are at ease in her presence. 
They know her voice and manner of speaking; and they more 
readily understand her than they would another. In these 
matters they are placed at a disadvantage when examined by a 
stranger. 

We should expect this disadvantage to be most evident in 
the lower classes; and an inspection of the medians of Table 
XXVI shows how strikingly this is true. The children of the 
3d grade fell off 10.5% ; of the 4th, 9% ; of the 5th, 6th, and 7th 
approximately 5.5%; and of the 8th only 2.5%. The force, 
whatever its nature, tending to depress the results at schools VI 
and VII was clearly operative to a greater degree in the lower 
classes and to a much less degree in higher classes. The effect of 
a change from the class teacher to a stranger as examiner would 
be expected to bring about results of just such a nature. 

But the obvious advantage of having the same examiner for 
every class (in this case there were two working together) is that 
however the results in general may be lowered, there is a better 
opportunity to compare class with class or grade with grade, 
or school with school. 

The real reason why this supplementary testing was under- 
taken was to verify the median intervals that had been derived 
from the original testing. The fact of classes being examined 
by the same persons is of great value for this purpose. If, with 
this factor of the examiner made constant, we find that these 
median distances in spite of a falling off of grade performances 
remain substantially the same, we shall be justified in feeling that 
our former results are reasonably reliable. 



68 Spelling Ability — Its Measurement and Distribution 

TABLE XXVII 

Number and Per Cent of Pupils in Each Grade Whose Ability Equalled 

or Exceeded that of the Median Pupil in Every Other Grade, 

with the P.E. Values Corresponding to Each Per Cent. 

Schools VI and VII Combined with Schools II, III, 

IV and V 







3d 
Grade 


4th 

Grade 


5th 
Grade 


6th 
Grade 


7th 
Grade 


8th 
Grade 


3d grade. . . 
N=749.... 


No. 

% 
P.E. 




110 
14.7 
1.55 


38 
5.1 
2.425 


12 
1.6 
3.18 


4 
.5 

3.82 





? 


4th grade. . 
N=763.... 


No. 

% 
P.E. 


619 
81.1 

—1.31 




223 

29.2 
.81 


90 
11.8 
1.76 


45 
5.9 
2.32 


16 
2.1 
3.01 


5th grade . . 

N=805. . . . 


No. 

% 
P.E. 


758 

94.2 
—2.33 


584 
72.5 
—.89 




227 

28.2 
.86 


121 
15.0 
1.54 


47 
5.8 
2.33 


6th grade. . 
N=668. . . . 


No. 

% 
P.E. 


655 

98.1 
—3.08 


597 

89.4 
—1.85 


511 

76.5 
—1.07 




241 
36.1 
.55 


112 
16.8 
1.43 


7th grade. . 
N=702.... 


No. 

% 
P.E. 


698 

99.43 
—3.75 


678 

96.6 
—2.71 


615 

87.6 
—1.71 


482 
68.7 

— .72 




188 
26.8 
.92 


8th grade. . 
N=570.... 


No. 

% 
P.E. 


570 
100.0 
? 


565 

99.12 
—3.52 


544 

95.44 
—2.50 


502 

88.1 
—1.75 


428 

75.1 
—1.00 





In Table XXVII we give the number and per cent of pupils 
who equal or surpass the medians of other grades than their 
own with the corresponding P.E. values. In this table are com- 
bined the children who were examined at schools VI and VII 
with those examined at schools II, III, IV and V. It is to be 
compared with Table XV (page 36). Table XXVIII gives the 
median distances as derived from the P.E. values of Table 
XXVII. Compare with Table XVI (page 39). The average 
distances, 1.37, .87, .90, .66, and .86, are to be compared with 
the values for the same distances as derived on page 39, namely 
(correct to 2 decimal places) 1.35, .84, 1.05, .66, and .91. Only 
in the case of M 5 _ 6 is there an important difference. The average 
difference, including that of M 5 _ 6 , is .05 ; excluding it, the average 
difference is but .025. The entire range from M, to M 8 is found 



Arrangement of Words of List by Teachers' Judgments 69 

to be 4.66 as compared with 4.81 according to the primary 
testing. These correspondences are, we feel, quite close enough 
to establish the essential reliability of our original figures. 



TABLE XXVIII 
Median Distances Derived from the P.E. Values of Table XXVII 





^3-4 


^4-5 


^5-6 


^6-7 


M 7-8 


Average 


1.55 

(direct) 

1.61 

(M 3 _ 5 -M 4 _ 5 ) 

1.42 
(M 3 _ 6 — M 4 _ 6 ) 

1.50 

(M 3 _-M 4 _ 7 ) 

? 

(M 3 _ 8 —M 4 _ 8 ) 

1.31 

(direct) 

1.44 

(M 5 _ 3 -M 5 _ 4 ) 

1 23 

(M 6 _ 3 -M 6 _ 4 ) 

1.04 

(M 7 _3-M 7 _ 4 ) 

? 

(M 8 _ 3 — M 8 _ 4 ) 


.87 
(M 3 _ r ~M 3 _ 4 ) 

.81 
(direct) 

.90 
(M 4 _ 6 -M 5 _ 6 ) 

.78 
(M 4 _-ifef 5 _ 7 ) 

.68 

(M 4 _ 8 -M 5 _ 8 ) 

1.02 

(M 5 _ 3 -M 4 _ 3 ) 

.89 
(direct) 

.78 
(M 6 _ 4 -M 6 _ 5 ) 

1.00 

(M 7 _ 4 -M 7 _ 5 ) 

1.02 

(M 8 _ 4 ~M 8 _ 5 ) 


.76 

(M 3 _ 6 — M 3 _ 5 ) 

95 

(M 4 _ Q -M 4 _ 5 ) 

.86 
(direct) 

99 

(M 5 _ 7 -M 6 _ 7 ) 

.90 
(M 5 _ 8 — M 6 _ 8 ) 

.75 

(M 6 _ 3 — M 5 _ 3 ) 

.96 

(M 6 _ 4 -M 5 _ 4 ) 

1.07 
(direct) 

99 

(M 7 _ 5 -M 7 _ 6 ) 

.75 

(M 8 _ b ~ M 8 _ Q ) 


.64 

(M 3 _ 7 — M 3 _ Q ) 

.56 

(M 4 _ 7 ~M 4 _ 6 ) 

.68 

(M 5 _ 7 -M 5 _ 6 ) 

.55 

(direct) 

.51 

(M & _ 8 — M 7 _ 8 ) 

.67 

(M 7 _ 3 —M 6 _ 3 ) 

.86 

(M 7 _ 4 -M 6 _ 4 ) 

.64 
(M 7 _ 5 -M 6 _ 5 ) 

.72 
(direct) 

.75 

(M 8 _ 6 — M 8 _ 7 ) 


? 

(M 3 _ 8 — M 3 _ 7 ) 

.69 

(M 4 _ 8 -M 4 _ 7 ) 

.79 
(M 5 _ 8 -M 5 _ 7 ) 

88 
(M Q _ 8 —M Q _ 7 ) 

.92 
(direct) 

(M 8 _ 3 -M 7 _ 3 ) 

.81 
(M 8 _ 4 -M 7 _ 4 ) 

.79 

(M 8 _ 5 -M 7 _ 5 ) 

1.03 

(M 8 _ -itf 7 _ 6 ) 

1.00 

(direct) 


1.37 


.87 


.90 


.66 


.86 


§ 13 

A 
List 
scho 
othe 
resu 
judg 


7. Arranger} 

certain ordei 
having beer 
ols and to a 
r schools (T 
t with an a 
ment of teac 


lent of the 
Teachers 

• of difficulty 
i determined 
certain exte 
able XIX, a 
rrangement 
hers becomes 


Words of t) 
? Judgments 

of the 50 w< 

as the rest 

nt verified b; 

rid Fig. 20), 

;>f the same 

interesting. 


ie Preferred 

Drds of the I 

It of testing 

y a record f 

a compariso 

words basec 

It has an ir 


List by 

'referred 
• in five 
rom two 
n of the 
. on the 
nportant 



70 Spelling Ability — Its Measurement and Distribution 

bearing on the whole spelling situation because it is by individual 
judgment as to the difficulty of words that lists are made up 
and graded for classes, schools, or systems of schools. How 
far such grading is reliable may be gathered by finding out to 
what extent individual judgment squares with the results of 
actual testing. 

For this purpose the 50 words of the Preferred List were 
arranged alphabetically and distributed to a number of teachers. 
They were asked to arrange the words in what they judged to be 
their order of difficulty for children to spell, beginning with 
the easiest. They were particularly requested to do the work 
without consulting any one. Two hundred arrangements were 
secured. They differed widely — so widely that whatever may 
be the value of a consensus of many individuals, the trust- 
worthiness of the judgment of a single teacher appears to be 
almost of no value. It may be good and it may be bad; and it 
is about as likely to be the one as the other. With one notable 
exception, the agreement with the record was closer for the 
easiest and hardest words than for those of medium difficulty. 
This might have been expected from the fact that, as shown in 
Fig. 20, the words at the ends of the scale differ by larger 
amounts than do those nearer the middle. The exception 
referred to is the word " too," which, although it was in every 
school the hardest word to spell, was by more than a fourth 
of the teachers judged to be actually the easiest, or next to the 
easiest, in the list. The deviation of individual judgments from 
the record is shown by figures taken from a random sampling of 
the two hundred teachers' arrangements. Five such arrange- 
ments being taken by chance from the whole number, proved to 
be those of teachers No. 7, No. 88, No. 109, No. 134, and No. 
178. They ranked the 5th, 10th, 15th, 20th, .... 50th words 
(record) as follows: 

Record 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 

No. 7 22, 15, 1, 31, 34, 33, 38, 47, 14, 10 

No. 88 15, 9, 4, 41, 13, 24, 33, 35, 20, 1 

No. 109 9, 11, 18, 39, 35, 36, 22, 25, 29, 10 

No. 134 24, 22, 23, 20, 6, 44, 7, 42, 35, 11 

No. 178 21, 2, 7, 22, 14, 29, 49, 38, 28, 1 




h*J 



Arrangement of Words of List by Teachers' Judgments 7 1 

It is, however, to be expected that when a large number of 
judgments are taken together, wide disagreements with a true 
arrangement will tend to disappear, and a resultant will be 
obtained that may be expected to be closer to the facts than 
any single judgment would ever be likely to be. The statistical 
treatment of the 200 judgments was based on the theorem, 
" Differences that are equally often noticed are equal, unless the 
differences are either always or never noticed." It is an abbre- 
viation of the method used by Professor Thorndike in deriving his 
scale for Handwriting and by Dr. Hillegas in his similar work for 
English Composition. We have not felt the necessity of making 
comparisons of the judgment of each word with that of every 
other word, because the nature of our material has enabled us 
to derive our scale by a more direct method. Since we are here 
concerned with a comparison only, we have been content to 
proceed as follows : The record shows " only " to be easier 
than " even." What per cent of the individuals who arranged 
the words for difficulty so judged ? As between " even " and 
" smoke " ; " smoke " and " chicken " ; " chicken " and " front " ; 
etc., what per cent of the judgments indicate that the first word 
of each pair is easier than the second, as the record shows ? The 
following is found to be true for the first six words : 

"only" was judged easier than 

"even" 

"smoke" 

"chicken" " 

"front" 

Now when the difficulty of a word is judged by a very great 
number of judges some will overestimate its difficulty, others will 
underestimate it. Those who make small errors will be more 
numerous than those who make large errors. The frequency 
of the judgments will take the form of the curve of the proba- 
bility integral whose base-line represents the amounts of difficulty 
which the word in question is judged to have. The point on the 
base-line which corresponds to the greatest frequency of judg- 
ments represents the central tendency of the judges in rating 
the word. It is therefore the point which represents the difficulty 
of the word as determined by individual judgments. Two or 
more words may be compared for difficulty if we know the per 



even " 


by 38.5% of the judges. 


smoke " 


" 67.5% " " 


chicken " 


" 72.0% " " 


front " 


" 48.0% " " " ■ 


another " 


" 53-0% " - 



72 Spelling Ability — Its Measurement and Distribution 



cent of judges who rate one word easier (or harder) than the 
other. 

Fig. 22 shows the curves for the first six words arranged to 
show the per cent of " easier " judgments noted above. The 
curve for " even " is so placed that 38.5% of its area is to the 
right of YO — the median axis of the curve for " only " when that 

TABLE XXIX 

Comparison of Results by Teachers' Judgments and by the 

Record. Preferred List 



Word- 

Num- 

ber 

(scale) 


Word 


%of 

times 

each 

word was 

judged 

easier 

than the 

following 

word 


Rank 


Word- 
Num- 
ber 
(scale) 


Word 


% of 

times 

each 

word was 

judged 

easier 

than the 

following 

word 


Rank 


By 

Teach- 
ers' 
Judg- 
ments 


By 

the 
Re- 
cord 


By 

Teach- 
ers' 
Judg- 
ments 


By 

the 
Re- 
cord 


3 
1 
4 
28 
5 

9 

2 

8 

11 

16 

46 
41 
6 
45 
25 

49 
15 
50 
22 
21 

10 
48 
12 
35 

42 


only 

even 

smoke 

chicken .... 
front 

another . . . 

lesson 

bought. . . . 

pretty 

nails 

butcher . . . 
Tuesday. . . 

sure 

answer. . . . 
nor 

raise 

cousin .... 
beautiful . . 

touch 

freeze 

forty 

instead .... 

wear 

tailor 

tying 


38.5 
67.5 
72.0 
48.0 
53.0 

39.5 

86.0 
29.5 
27.5 
89.0 

65.5 

19.5 
74.5 
10.0 
90.0 

69.0 
53.0 
41.5 
29.5 
31.5 

75.0 
33.0 
44.0 
72.5 
55.0 


2 
1 
3 
7 
6 

8 

4 
18 
10 
11 

19 
29 
13 

21.5 
5 

21.5 

37 

39 

33 

20 

12 
24 
17 
14 
25.5 


1 
2 
3 
4 
5 

6 
7 
8 

9 • 
10 

11 
12 
13 
14 
15.5 

15.5 

17.5 

17.5 

19 

20 

21.5 

21.5 

23 

24.5 

24.5 


14 
7 
40 
38 
43 

13 
17 
33 
44 
30 

20 
23 
19 

24 
47 

37 
29 
36 

18 
34 

26 
32 

27 
31 
39 


minute. . . . 

pear 

towel 

tobacco . . . 
whole 

button .... 
janitor .... 
quarrel. . . . 
against. . . . 
circus 

sword 

whistle. . . . 
stopping... 
carriage . . . 
guess 

telephone. . 
choose .... 
telegram . . . 

saucer 

saucy 

already 

pigeons .... 
beginning. . 

grease 

too 


25.5 
51.0 
69.0 
56.0 
15.5 

79.5 

81.0 
39.5 
43.5 
77.0 

44.5 
21.5 
87.0 
29.0 
54.0 

27.0 

56.0 
54.5 
58.0 
35.5 

91.5 

16.0 
64.0 
21.0 


27 
15 
16 
25.5 

28 

9 
23 
44 
40 
35 

48 
46 
30 
49 
43 

45 
31 
36 
38 
41 

34 
50 

42 
47 
32 


26 
27 
28 
29 
30 

31 
32 
33 

34 
35 

36. 

36.5 

38 

39 

40 

41 
42 
43 
44 
45 

46 
47 
48 
49 
50 



axis is produced; 67.5% of the curve for "smoke" is to the 
right of Y 1 1 produced; J2% of the curve for " chicken " is to 
the right of Y 2 2 produced; and so on. H O 1 is the difference 
in difficulty between " only " and " even " ; K O 2 , between 
" even " and " smoke " ; P s , between " smoke " and " chicken " ; 
S O 4 , between " chicken " and " front " ; and T O 5 , between 
" front " and " another." When less than 50% of the judges 
regard the first of a pair of words as easier, the second is, of 



Arrangement of Words of List by Teachers' Judgments 73 

Y 




Fig. 22. Diagram showing difference in difficulty between words by- 
Teachers' Judgments. 



74 Spelling Ability — Its Measurement and Distribution 

course, judged the easier, and the difference in difficulty is a nega- 
tive one. Such is the case in the only-even and chicken-front 
pairs. The scaling of these six words is diagrammatically shown 
in Fig. 22 by producing each median vertical to meet the line AB. 
It will be seen that the order of difficulty is not the same as 
the order obtained by testing. 

Table XXIX shows for the entire 50 words the per cent of 
teachers who judged each word easier than the following word 
and the rank of each word for difficulty by record and by 
teachers' judgments. In spite of the fact that the opinion of 
single teachers is so unreliable, the combined judgments of a 
group as large as 200 yield an arrangement which agrees closely 
enough with the arrangement by record to confirm and support 
the latter in no small degree. The correlation, by the ' foot-rule ' 
method, is found to be 0.79, which may quite properly be 
regarded as satisfactory. 

It may be worth while to point out, however, that in practice 
the selection and arrangement of words for teaching are not the 
work of a large number of individuals. These things are usually 
done by a single teacher for a class or by a text-book writer for 
as many classes as use his book. Not two hundred, probably not 
even ten, persons judge as to the selection and arrangement of 
the words in the lists now used in most schools. The length of 
such lists, moreover, would seem to preclude the possibility of 
a satisfactory judgment as to difficulty by individuals. Probably 
if our own list of 50 words had been shorter, the teachers would 
have worked more accurately. The several thousand words in 
a spelling-book certainly constitute a list about which there may 
be expected to be wide and numerous disagreements. We alluded 
in Section 3 above to our attempt to secure for school use a 
5000-word vocabulary graded by years and based upon the agree- 
ments of five spelling-books. This task proved to be very difficult 
precisely because of the total absence of agreement as to grading 
in the case of hundreds of words. One speller would assign 
words to the third grade which another would put in the sixth, 
seventh, or eighth. Gradings three, four, and even five years 
apart occurred with remarkable frequency. 

The obvious way (and the necessary way, it would seem) to 
grade words for difficulty is not by some one's opinion of how 



Rice Sentence Test. Easy 50-Word Test 75 

hard they are, but by actually " trying them out." In matters of 
handwriting and composition the judgment of individuals is all- 
important, because merit in either is precisely a matter of judg- 
ment. One sample is better than another only because competent 
persons think so. On the contrary, one word is harder to spell 
than another not because we think so, but because more people 
misspell the one than the other, or because it takes more time to 
learn to spell the one than the other. It is strange, therefore, 
that no spelling-book has yet appeared based upon a study of 
how frequently children misspell the words of which it is com- 
posed. In fact no study of spelling, that we know of, has done 
more than obtain individual ratings of pupils, based on the tacit 
assumption that the words used are all equally difficult to spell. 
No investigation has been thought necessary of the words them- 
selves. The results of this section, although by no means 
thoroughly worked out, sufficiently indicate the present unrelia- 
bility of individual judgment with regard to words, unless the list 
is very short and the judgments very numerous. It is quite 
possible that at some later time, after studies of words based on 
actual tests have been frequently made, our judgment of word 
difficulties may be greatly improved. Our opinion as to how hard 
words are might then become a valuable supplement to the 
conclusions of the investigator. 

§ 18. Rice Sentence Test. Easy 50-Word Test 

During the middle week of the fall term of 19 12, the sentence 
test used by Rice — and afterwards by Cornman — was dictated 
to 1984 pupils in schools II, III, and VIII. Children of the 4th 
and 5th grades wrote sentences containing 50 words ; those of the 
6th, 7th, and 8th grades wrote 41 of the same words together with 
36 additional words — JJ in all. The entire test follows : 

Rice Sentence Test 

While running he slipped. I listened to his queer speech, but 
I did not believe any of it. The weather is changeable. His loud 
whistling frightened me. He is always changing his mind. His 



76 Spelling Ability — Its Measurement and Distribution 

chain was loose. She was baking cake. I have a piece of it. Did 
you receive my letter? I heard the laughter in the distance. 
Why did you choose that strange picture? ^Because I thought 
I liked it. It is my purpose to learn. Did you /os^ your almanac? 
I gave it to my neighbor. *I was writing in my language book. 
Some children are not careful enough. Was it necessary to keep 
me waiting so long? Do not disappoint me so o/few. I have 
covered the mixture. He is getting better. *A feather is Zi^/jf. 
Do not deceive me. I am driving a new horse. *Is the surface 
of your desk rough or smooth? The children were hopping. 
This is certainly true. I was very grateful for my elegant present. 
If we have patience we will succeed. He met with a severe 
accident. Sometimes children are not sensible. You had no 
business to answer him. You are not sweeping properly. Your 
reading shows improvement. The ride was very fatiguing. I am 
very anxious to hear the news. I appreciate your kindness I 
assure you. I cannot imagine a more peculiar character. I 
guarantee the book will meet with your approval. Intelligent 
persons learn by experience. The peach is delicious. I realize 
the importance of the occasion: Every rule has exceptions. He is 
thoroughly conscientious; therefore I do trust him. The elevator 
is ascending. Too much praise is not wholesome. 



TABLE XXX 
Distribution of Individual Ratings. Rice Sentence Test. 



Per Cent Correct 


4th Grade 


5th Grade 


6th Grade 


7th Grade 


8th Grade 


No. 


% 


No. 


% 


No. 


% 


No. 


% 


No. 


% 


0-10 

11-20 

21-30 


38 

54 
61 


7.5 
10.7 
12.0 
10.8 
14.2 
14.6 
11.6 
10.7 
6.1 
1.8 


3 
13 
18 
49 
62 
76 
76 
91 
65 
17 


.6 

2.8 

3.8 

10.4 

13.2 

16.2 

16.2 

19.4 

13.8 

3.6 


2 
13 
21 
46 
53 
80 
70 
62 
40 

9 


.5 

3.3 

5.3 

11.6 

13.4 

20.2 

17.7 

15.7 

10.1 

2.3 



3 
5 
20 
32 
52 
53 
77 
95 
30 




.8 

1.4 

5.4 

8.7 

14.2 

14.4 

21.0 

25.9 

8.2 










2 

15 

37 

61 

96 

33 







31- 40 


55 

72 
74 
59 
54 
31 
9 





41- 50 


.8 


51- 60 


6.1 


61- 70 


15.2 


71- 80 


25.0 


81- 90 


39.3 


91-100 


13.5 






Totals 

Medians 


507 


48.17 


470 


64.13 


396 


58.57 


367 


73.86 


244 


82.07 



Rice Sentence Test. Easy 50-Word Test 



77 



The 4th- and 5th-year test ends with " This is certainly true." 
The test for the upper grades comprises all the sentences except 
the four marked with an asterisk. The test words are italicized. 

The principal object in giving this test was to obtain scores 
for a new series of words by which the grade and general scales 
(Figs. 14-20) could be supplemented and extended. The papers, 
however, were also rated for individual performances. Subject 



/o 



O /0 2.0 30 H-O FO 60 70 SO ^0 I0O 



AS" 
/0 



/O £0 30 H-C SO 60 70 BO 70 /OO 

Fig. 23. Relative frequencies of different percentages correct, 4th grade; 

Rice Sentence Test; Table XXX. 
Fig. 24. Same as Fig. 23, but for 5th grade. 

to the limitations of regarding all words as equal, the results on 
this basis may be used to supplement those of Rice and Cornman. 
Rice gives nothing but grade averages, and Cornman gives the 
same for two tests after an interval of one year. We shall 
continue to use the median as a measure of central tendency and 
shall give a distribution of pupils' ratings for each grade. Table 
XXX gives the distribution by groups of ten with totals and 
medians. Fig. 23 and Fig. 24 show graphically the distribution 



78 Spelling Ability — Its Measurement and Distribution 

for the 4th and 5th grades. Fig. 8, Fig. 9, and Fig. 10 (page 
33) gi ye ^e same for 6th, 7th, and 8th grades. 

These medians indicate a performance for these schools poorer 
than Rice indicates for most of his schools and much poorer than 
Cornman's results for the two schools which he tested. We can- 
not account for this because neither of these investigators tells 
how he rated the pupils' papers. In our own testing omitted and 
illegible words were counted as wrong. It is probable that the 
manner in which Rice chose his schools would give him those in 
which better than average work was being done ; while Cornman's 
two schools were without doubt devoting an unusual amount of 
attention to spelling under his personal guidance. Such being 
the case, it is quite possible that the results here given more 
nearly approach typical conditions, than do those of either of 
these writers. 

TABLE XXXI 
Per Cent Correct for Each Word in Each Grade with Corres- 
ponding P.E. Values. Rice Sentence Test. See Fig. 25 



No. of 
Word 



7 

8 

9 

10 

11 
12 
13 
14 

15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
25 

26 
27 
28 
29 
30 



Word 



running 

slipped 

listened 

queer 

speech 

believe 

weather 
changeable. . 
whistling . . . 
frightened . . 

always 

changing . . . 

chain 

loose 

baking 

piece 

receive 

laughter. . . . 
distance. . . . 
choose 

strange 

picture 

because . . . . 

thought 

purpose 

learn 

lose 

almanac. . . . 
neighbor. . > . 
writing 



4th Grade 



% 



48.0 
30.0 
29.6 
56.9 
45.3 

37.2 
70.9 

27.7 
27.3 
17.8 

53.8 
58.5 
51.8 
24.7 
63.4 

58.5 
21.1 
59.9 
35.6 
41.7 

57.7 
69.6 
66.2 
58.7 
21.7 

70.6 
46.4 
10.1 
27.5 
56.3 



P.E. 



+ .074 

+ .777 

+ .794 

— .258 
+ .175 

+ .484 

— .816 
+ .878 
+ .895 
+ 1.368 

— .141 

— .318 

— .067 
+ 1.014 

— .508 

— .318 
+ 1.191 

— .372 
+ .547 
+ .311 

— .288 

— .761 

— .620 

— .326 
+ 1.160 

— .803 
+ .134 
+ 1.892 
+ .8 

— .235 



5th Grade 



66.0 
34.8 
40.4 
58.8 
41.4 

49.7 
57.5 
31.3 
40.0 
42.7 

68.8 
69.2 
59.8 
49.1 
75.7 

62.2 
51.7 
71.4 
67.2 
46.3 

74.2 
87.5 
83.9 
72.4 
47.3 

84.9 
53.1 
21.5 
66.6 
74.0 



P.E. 



— .612 
+ .579 
+ .360 

— .330 
+ .322 

+ .011 

— .280 
+ .723 
+ .376 
+ .273 

— .727 

— .744 

— .368 
+ .033 
—1.033 

— .461 

— .063 

— .838 

— .660 
+ .138 

— .963 
— 1 . 706 
—1.459 

— .882 
+ .100 

—1.531 

— .119 
+ 1.170 

— .636 

— .954 



6th Grade 



% 



76.8 
42.9 
53.5 
77.3 
72.0 

64.4 
82.8 
46.7 
49.0 
55.6 

78.5 
74.5 
75.3 

45.2 
83.6 

69.9 

59.8 
75.5 

75.8 
56.8 

86.9 
94.4 



66.9 

93.2 
56.8 
38.6 
65.2 



P.E. 



+ .265 

— .130 
—1.110 

— .864 

— .547 
—1.403 
+ .123 
+ .037 

— .209 

— 1 . 170 

— .974 
—1.014 
+ .179 
—1.450 

— .773 

— .368 
—1.024 
—1.038 

— .254 

— 1 . 663 
—2.357 



— .648 

—2.211 

— .254 
+ .430 

— .579 



7th Grade 



% 



85.0 
51.8 
69.8 
79.0 
77.1 

62.1 

88.0 
66.8 
68.7 
71.4 



88.0 
63.2 
93.5 

83.7 
62.1 
88.8 
88.0 
83.1 

93.5 
97.5 



74.7 

95.9 
60.0 
58.6 
85.0 



P.E. 



-1 . 537 

- .067 

- .769 
-1 . 196 
-1.101 

- .457 
-1 . 742 

- .604 

- .723 

- .838 

-1 . 788 
-1.867 
-1 . 742 

- .500 
-2.245 

-1.456 

- .457 
-1.803 
-1 . 742 
-1.421 

-2.245 
-2.905 



-2.579 

- .376 

- .322 
-1 . 537 



8th Grade 



93.4 
70.9 
86.9 
87.3 
80.7 

76.6 
92.2 
65.6 
74.2 

85.7 

95.5 
91.4 
95.9 
81.6 
97.5 

90.6 

80.7 
96.3 
97.5 

85.7 

92.6 
98.8 



92.6 

99.6 
55.7 
72.1 
93.4 



P.E. 



-2 . 234 

- .816 
-1 . 663 
-1 . 692 
-1 . 286 

-1.076 
-2.103 

- .596 

- .963 
-1.582 

-2.514 
-2.035 
-2.579 
-1.335 
-2.905 



953 

286 
648 
905 

582 



-2.145 
-3.346 



—2.145 

—3.938 

— .213 

— .869 
—2.234 



Rice Sentence Test. Easy 50-Word Test 



79 



TABLE XXXI 

(Continued) 



No. of 
Word 



31 
32 
33 
34 
35 

36 

37 
38 
39 
40 

41 
42 
43 
44 
45 

46 
47 
48 
49 
50 

51 
52 
53 
54 
55 



61 
62 
63 
64 
65 



67 
68 
69 
70 

71 
72 
73 
74 
75 

76 
77 
78 
79 
80 

81 
82 
83 
84 
85 



Word 



language . 
careful . . . 
enough . . . 
necessary, 
waiting. . . 

disappoint 
often 

covered . . 
mixture . . 
getting. . . 

better. . . . 
feather. . . 

light 

deceive. . . 
driving . . . 

surface. . . 
rough. . . . 
smooth. . . 
hopping. . 
certainly . 

grateful . . 
elegant . . . 
present . . . 
patience. . 
succeed . . 

severe. . . . 
accident . . 
sometimes 
sensible . . 
business. . 



4th Grade 



sweeping . 
properly. . 
improvement 
fatiguing . . . 

anxious ... 
appreciate. 

assure 

imagine . . . 
peculiar ... 

character. . 
guarantee . . 
approval . . 
intelligent . 
experience. 

delicious . . . 
realize .... 
importance 
occasion. . . 
exceptions. 

thoroughly. . 
conscientious 
therefore . . 
ascending . . 

praise 

wholesome. 



40.3 
54.3 
54.9 
4.5 
55.9 

11.7 
51.6 
42.1 
33.6 
57.5 

80.6 
77.1 
77.5 
18.4 
59.7 

48.4 
64.2 
47.2 
58.1 
16.8 



P.E. 



+ .364 

— .160 

— .183 
+ 2.514 

— .220 

+ 1.757 

— .059 
+ .296 

+ .628 

— .280 

—1.279 
—1.101 
— 1 . 120 
+ 1.335 

— .364 

+ .059 

— .539 
.104 
.303 

1.427 



>th Grade 



% 



62.8 
58.6 


21.5 
66.8 

27.4 
57.5 
62.6 
62.6 

74.4 

91.8 
84.1 
90.5 
46.3 
77.1 

79.1 
69.8 
51.3 
58.1 
36.0 



P.E. 



— .484 

— .322 

— .693 

+ 1.170 

— .604 

4- .891 

— .280 

— .476 
.476 

— .972 

—2.064 
—1.481 
—1 . 944 
+ .138 
—1 . 101 

—1.201 

— .769 

— .048 

— .303 
+ .531 



% 



3 

42.7 
82.3 

34.6 

75.8 
77.5 
83.3 

87.4 



6th Grade 



68.4 
57.1 

39.1 
53.5 
69.7 
43.4 
53.0 

40.9 
45.5 
52.5 
34.3 
46.0 

74.0 
87.4 
61.1 
59.6 
12.6 

49.0 

31 

58.1 

33.6 

24.0 

40.2 

11. 

38.1 

37.1 

44.4 

31.3 
53.5 

47.5 
34 

4S 



IS 



36 
37 
69.0 
56.3 



P.E. 



7th Grade 



% 



731 85.8 



— 1 . 264 
+ .273 
—1.374 

+ .588 
—1.038 
— 1 . 120 
— 1 . 432 
— 1 . 699 

—2.425 



- .130 
-1 . 749 



— .710 

— .265 

+ .410 

— .130 

— .765 
+ .246 

— .112 

+ .341 

+ .168 

— .093 
+ .600 
+ .149 

— .954 

— i.e 

— .418 

— .360 

+i.e 

+ .037 
+ .702 

— .303 
+ .628 
+ 1.047 

+ .368 
+ 1.772 
+ .449 
+ .448 
+ .209 

+ .723 

— .130 
+ .093 
+ .579 
+ .067 

+ 1.318 
+ 4.167 
+ .516 
+ .468 

— .735 

— .235 



91.0 
37.6 
89.9 

32.4 
87.2 
90.2 
91.0 
94.6 

98.6 



54.8 
65.7 



81.2 
79.0 

58 
65 

79.0 
63.0 
70.8 

61.3 

68 

67.3 

55.0 

53.7 

86.9 
92.1 
73.0 
69.5 
25.3 

66.2 

49.0 

68.9 

47 

46.3 

47.1 
19.9 
56.9 
43. 



P.E. 



bcs.o 
61.6 


65.7 


73.3 


44.4 


57.2 


31.1 


1.6 


62.4 


52.0 


78.2 


74.7 



—1 . 589 
—1 . 988 
+ .468 
—1 . 892 

+ .657 
—1 . 685 
—1.918 
—1 . 988 
—2.384 

-3.258 



— .179 

2.177 



-1.313 
-1.196 

- .322 

- .600 
-1.1 

- .492 

- .812 

- .426 

- .731 

- .665 

- .187 

- .138 

-1 . 663 
-2.093 

- .909 

- .756 



— .620 
+ .037 

— .731 
+ .085 
+ .138 

+ .108 
+ 1.253 
— 1 . 258 
+ .239 

— .512 

— .437 

— .600 

— .922 
+ .209 

— .269 

+ .731 
+ 3.171 

— .468 

— .074 
— 1 . U 



8th Grade 



% 



.1 
98.4 
61.5 
92.2 

38.9 
92.2 
97.1 
97.1 
97.5 

100.0 



79.5 

8 



.3 

91.0 

61. 

69.3 

91.4 

.7 

.7 

70.9 

85.2 
82.8 
65.2 
68.4 

93.4 
94.7 
86.5 
86.5 
31.1 

84.0 

74 

86.1 

66.4 

56.1 

78.7 
25.8 
75.4 
50.4 
68.9 

85.2 

73.4 

81.6 

49 

76.2 

53.3 

19.7 

79.9 

55 

95.9 

86.1 



P.E. 



-1 . 749 
-3 . 182 
- .434 
-2 . 103 

.418 

-2 . 103 

-2.811 

—2.811 

2.905 



— 1 . 222 
—3 . 346 



1.843 
—1.988 

.449 

.748 

2.035 

—1.286 

1.286 

. 816 

— 1 . 549 

1 . 403 

.579 

— .710 

—2.234 
—2.397 
— 1 . 636 
— 1 . 636 
+ .731- 

—1.475 

— .982 
— 1 . 609 

— .628 

— .228 

—1.181 
+ .963 
—1.019 

— .015 

— .731 

— 1 . 549 

— .927 
—1.335 
+ .015 
—1.057 

— .123 
+ 1.264 
— 1 . 243 

— .213 
—2 . 579 
— 1 . 609 



80 Spelling Ability — Its Measurement and Distribution 

With respect to the ratings of words, Table XXXI gives for 
each grade the per cent of correct spellings and the P.E. values 
calculated from the grade medians, assuming a normal distribu- 
tion. Fig. 25 (insert) shows the lines the same words arranged 
on a linear scale for grades 4, 5, 6, 7, and 8. Above the lines 
the arrangement of the words of the Preferred List is given. This 
latter is a repetition of the scales of Figures 15, 16, 17, 18 and 
19 (p. 44). It will at once be seen that the former scales, 
obtained by using the Preferred List, have been filled in and have 
been extended much further to the right. 

Just as the more difficult words of the Rice Test may be used 
to extend the scales to the right, so the easier words of the Easy 
50- Word Test may be used to extend it towards the left in certain 
grades. In the grades of the second school year the latter were 
the only words used. Although the primary object in giving the 
Easy 50- Word Test was to enable us to give a position to the 
zero-point, and although for this purpose the ratings of individual 
pupils were sufficient, nevertheless the per cent of correct spellings 
for each word in each grade. (2a, 2b, 3d, and 4th) was also 
calculated. 

Table XXXII gives these per cents and the corresponding P.E. 
values. It will be noted that there are six words {even, only, 
pretty, sure, touch, and front) that are common to this list and 
to the Preferred List. 

TABLE XXXII 

Per Cent Correct for Each Word in Each Grade with Corres- 
ponding P.E. Values. Easy 50-word Test 



No. 

of 

Word 


Word 


2a Grade 
175 Pupils 


26 Grade 
169 Pupils 


3d Grade 
168 Pupils 


4th Grade 
316 Pupils 


% 


P.E. 


% 


P.E. 


% 


P.E. 


% 


P.E. 


1 
2 
3 
4 
5 

6 

7 

8 

9 

10 


you 

will 

hear 

him 

coming . . 

he 

is 

on 

the 

road .... 


52.6 
46.9 
37.1 
25.7 
9.7 

62.3 
65.7 

57.7 

65.7 

4.6 


— .097 
+ .119 
+ .489 
+ .968 
+ 1.926 

— .464 

— .600 

— .288 

— .600 
+2.498 


71.6 
83.4 
53.3 
71.6 
33.7 

88.7 
94.7 
88.7 
94.7 
20.7 


— .847 
— 1 . 438 

— .123 

— .847 
+ .624 

—1.795 
—2.397 
—1.795 
—2.397 
+ 1.211 


83.9 
96.4 
60.1 
81.0 
57.1 

98.2 
96.4 
94.6 
94.6 
53.6 


—1.459 
—2.667 

— .380 
— 1 . 302 

— .265 

—3.111 

—2.667 
—2.384 
—2.384 

— .134 


93.4 
99.1 
79.4 
97.5 
83.5 

99.7 
98.4 
97.8 
97.8 
82.6 


—2.234 
—3.506 
—1.217 
—2.905 
— 1 . 444 

-^.083 
—3.182 
—2.986 
—2.986 
—1.391 



+ 60 *80 WOO +i2Q iiH-0 *IL0 +/SO +200 






"Lvsv 



v5~ 4voAe., 



i ,i i_ 



50 2 8 & 28 

3* 



J/ 

J llll I Li_ 



/GcrftAe. 



J3 &4 ^7* 76 8 1 CS 72 

n 



7 £o-ff.cLe-. 



3C Ji 



3 Gcvcv^je^. 



S2 



sy 50-Word List, Table XXXII. These scales are not absolutely 
:he Rice List at +317. For the lists referred to see Appendix II. 






-3iO -UO -J'/O -320 300 -280 - ZLO -240 -220 200 -/80 -MO - 



140 -J20 -100 -80 -(,0 -4Q -20 o * 20 * /,0 t 



40 + 60 t 80 '/CIO .120 i/VO </£0 '/SO • ZOO 



~?Te<>««A."TJwY 



TW Senfenu- TmV- 



i«2i town *ow« j. 



Jl «JH*J3^ JJ 



-^Gsro-cU 



fcoa^ So-WoyaIaV* 



XW**tA."LlA 



28 f M 



\vce StAewt-c '"?e!,V 



S^&taA*.. 



i& * '<» V ir *J w J J ' J0 * 8 



^Wv^T-'^V 



"■Hoioll liM 3331,1 



Hv» Se 



,7«- •» /; ,r /f, 






?*J,«,«\Xv»Vr 



J. * 



7*&v««u. 



g S^Yfeou. v \e>V 



J .' 3 V 2 ,0 7. 3 '« «•,.' "* *> t.v s £ 



ftunrti*: 



* *» 22 /* h m ft + 7 K Ji Ji 



TWW.ac TtsV. 



J8 /« « // & 



,71, 177 2t 2 «t 



10 , I ompleted Grade Scale. Grades i to 8. Combining the Rice Sentence List, Table XXXI, with the Preferred List, Table XVII, to which is also added, for the 4th grade, th< Easj ;o Word 1 1 1. I able XXXII These scales are nol ,1 In 
nplete. [he Jth grade scale should have word No. 34 of the Rice List at +255; the 6th grade scale should have word 82 of the Rice List at +415; the 7th grade scali I d havi 8 oi tin Rici List al | |i; F01 th< rred I > ppendi II 



Rice Sentence Test. Easy 30-Word Test 



81 



TABLE XXXII 

(Continued) 



No. 

of 

Word 



11 
12 
13 
14 
15 

16 
17 

18 
19 
20 

21 
22 
23 
24 
25 

26 

27 
28 
29 
30 

31 
32 
33 
34 
35 

36 
37 
38 
39 
40 

41 
42 
43 
44 
45 

46 

47 
48 
49 
50 



Word 



and .... 
almost . . 
sure. . . . 
to 

pass. . . . 

in 

front . . . 

of 

me 

I 

send . . . 

for 

every. . . 

day 

go 

into. . . . 
school. . 
but .... 

do 

not .... 

touch . . 
table . . . 
also .... 
has .... 
only. . . . 

one .... 
pair. . . . 
shoes. . . 
they . . . 
are 

at 

aU 

pretty. . 

no 

man 

ought . . 
steal . . . 
even . . . 

a 

penny. . 



2a Grade 


175 Pupils 


% 


P.E. 


51.4 


— .052 


.6 


+ 3.725 


1.1 


+3.392 


48.0 


+ .074 


1.1 


+3.392 


45.7 


+ .160 


12.0 


+ 1.742 


25.7 


+ .968 


54.3 


— .160 


45.1 


+ .183 


22.3 


+ 1.130 


34.3 


+ .600 


3.4 


+2.706 


36.6 


+ .508 


45.1 


+ .183 


34.3 


+ .600 


16.0 


+ 1.475 


13.1 


+2.767 


40.1 


+ .372 


33.7 


+ .624 


.6 


+3.725 


1.7 


+ 3.146 


2.9 


+2.811 


36.0 


+ .531 


5.7 


+2.344 


34.9 


+ .575 


2.9 


+2.811 


1.7 


+3.146 


16.6 


+ 1.438 


18.3 


+ 1.340 


43.4 


+ .246 


36.6 


+ .508 


2.9 


+2.811 


23.4 


+ 1.076 


50.9 


— .033 


1.1 


+3.392 


1.7 


+ 3.146 


5.1 


+2.425 


60.6 


— .399 


3.4 


+2.706 



2b Grade 
169 Pupils 



% 



85.2 

6.5 
5.9 

78.7 
27.8 

91.1 

27.8 
64.5 
84.6 
93.5 

25.4 

78.7 
39.6 
89.3 
87.0 

71.0 

60.4 
58.0 
74.0 
69.2 

4.7 
13.0 
15.4 
65.1 
16.6 

71.0 
11.2 

18.9 
34.3 
58.0 

72.8 
62.1 
20.1 
71.0 
74.0 



P.E. 



— 1 . 549 
+2.245 
+2.321 
— 1 . 181 

+ .873 

—1.997 

+ .873 

— .551 
— 1 . 512 

—2.245 

+ .982 
— 1 . 181 
+ .391 
—1.843 
—1.670 

— .820 

— .371 

— .299 

— .954 

— .744 

+2.483 
+ 1.670 
+ 1.512 

— .575 

+ 1.438 

— .820 
+ 1.803 
+ 1.307 
+ .600 

— .299 



+ 1 



900 
457 
243 
820 
954 



12.4 +1.713 
12.4 +1.713 
24.3+1.033 
72.8— .900 
13.6+1.629 



3d Grade 
168 Pupils 



% 



P.E. 



97.0—2.789 
42.9 + .265 
48.8 + .044 
93.5!— 2.245 
51.2— .044 



92.9 
48.2 
82.1 
88.7 
95.2 

48.2 
93.5 
66.7 
94.0 
89.3 

76.2 
81.5 
83.3 
85.1 
86.9 

29.8 
53.6 
35.7 
69.6 
53.0 

82.1 
42.9 
46.4 
64.3 
84.5 

75.0 
73.2 
43.5 
85.1 
79.8 

20.8 
29.8 
48.8 
86.3 
35.7 



—2.177 
+ .067 
—1.363 
— 1 . 795 

—2.468 

+ .067 
—2.245 

— .640 
—2.305 
—1.843 

—1.057 

—1 . 329 
—1.432 
—1.543 
—1.663 

+ .786 

— .239 
+ .543 

— .761 

— .112 

—1.363 

+ .265 
+ .134 

— .543 
—1.506 

—1.000 

— .918 
+ .243 
—1 . 543 
—1.238 

+ 1.206 

+ .786 
+ .044 
—1 . 622 
+ .543 



4th Grade 
316 Pupils 



% 



98.4 
56.3 
61.1 
94.9 
65.8 

93.0 
67.1 
91.5 
95.9 
100.0 

76.6 

96.8 

88.0 

100.0 

98.1 

89.9 
92.7 
98.4 
97.2 
96.8 

54.4 
94.3 
68.4 
91.5 
70.6 

94.6 

77.5 
76.9 
88.0 
95.9 

86.7 
90.2 
78.5 
96.5 
97.2 

53.8 
67.1 
67.4 
98.4 
63.6 



P.E. 



—3.182 

— .235 

— .418 
—2.425 

— .603 

—2.188 

— .656 
—2.035 
—2.579 

? 

—1.076 

—2.746 
—1.742 

? 
—3.077 

—1.892 
—2.155 
—3.182 
—2.155 
—2.746 

— .164 
—2.344 

— .710 
—2.035 

— .803" 

—2 . 384 
— 1 . 120 
—1.091 
— 1 . 742 
—2.579 

—1.649 
—1.918 
—1.170 
—2.686 
—2.155 

— .141 

— .656 

— .669 
—3.182 

— .516 



82 Spelling Ability — Its Measurement and Distribution 

Figures 26, 27, and 28 give the scales for these words. In 
Figure 2J it is indicated below the line with the omission of the 
six words noted in the last paragraph. Above the line the 
words of the Preferred List are reproduced from Figure 14. 
Since the Easy 50- Word Test was also given to 4th-grade chil- 
dren it is likewise scaled for that grade omitting the same six 
words. (Fig. 25, 4th grade, lower line.) 

We have, therefore, scales for every grade from the first half 
of the second grade to and including the eighth. All of these 
scales above the 2d grade are much richer than were those 
given in Section 12. There are fewer gaps in them and their 
range is greater. They may be used to great advantage in 
testing the spelling ability of children in any grade of the 
elementary school in which children are supposed to have any 
such ability. If it is not convenient to use a whole scale, certain 
words differing in difficulty by approximately equal amounts 
may be selected. Groups of words may be made each of equal 
difficulty as a group, or each differing from the preceding group 
by a fixed amount. The position of each word shows the weight 
which ought to be assigned to it for test purposes. 

Each of these grade scales refers to the median of the grade 
as the zero-point. In Figure 29 is shown a scale for all grades 
referring, as in Figure 20, to the median of the 3d grade as the 
zero-point. Above the line is shown the Preferred List as in 
Fig 20. Below it are arranged the words of Rice's Test; and 
on a parallel scale the Easy 50- Word List. Caution, however, 
ought to be observed in accepting too literally the showing of 
the last two lists. Rice's Test was not given to the 3d grade, 
and the Easy 50- Word Test was given to the 2d grade and 
was not given above the 4th. They cannot, therefore, be closely 
compared with the Preferred List. The effect of high grades 
is to make the words harder, of low grades to make them easier. 
In the case of the Rice Test the words are probably a little — 
but only a little — too far to the right, — i.e., farther toward the 
high end than they would have been had they been used in the 
3d grade — as Cornman used them. In the case of the Easy 
50- Word Test the words would be a great deal too far to the 
left if set down as the record indicated. The six words common 



+ 220 +-240 + 



3* & to 



IZ 13 31 



■IOQ i-lZQ 



is- 
4*v* J» n Hi Mm 



610 7/0 
220 +240 + 



, 30 *« 



5.V 7 ^7 41 1 



tt ^ 2f ij W 2d 



st being at +685 fi 



. 2 20 -^00 - 180 'I* ~ lJ *-° ~ 120 ~ 100 ~ S0 ~t° ~ lh0 ' 20 ° ' 20 + *° +Ca + 80 H ' 00 +/ ^n +IW t/6,0 + 180 + 200 +220 + 240 + 



260 + 230 +300 +320 + 340 + 36,0 -ri 



7 6 *, 



WJI 40 J? 2 



u lit is ii n z f % i *i 

320 -300 -280 --^0 -24° -220 -200 



a> >/« M. 27 2$ 






1^.21 4°" Gcvo-i-e.. 



-/40 -14-0 -120 -100 



TrAevveA-XUt 



do - +o -2a o +20 +40 +60 +3o +/oo f/20 +/40 +/6o +/80 +200 +220 



<r n 1, kn> hi"- a jf*? a 



I M i|iT>-\M«-A."W r 



^S7? J0*f 2?i /26 /8,4 //J" 



J JJ1 /0 IS 



-320 -300 -280 -260 -240 -220 -200 -/80 -/60 -19-0 -/20 -/00 -80 -60 -40 -20 +20 +40 + 60 



21s 2/ /o 3^7 a 

\ic, 2.8 GcenevcA ScnU Vov GwoAm 2aVoM- 



2X0 270 2?0 3/0 J30 o'^J J70 390 410 430 4S~0 >i70 490 SJO S30 356 S70 S~9~0 no 630 6S~0 670 690 710 730 750 710 790 8J0 %30 &50 g.70 Mo 9/0 930 W ?70 110 /010 
UO -220 -200 -/80 -ICO -140 - IZO - 100 -80 -CO - 40 -20 + 20 +40 +60 + SO -woo +/20 +140 +160 +/80 +200 +220 +240 + 2£0 +280 +300 +320 +M0 +360 +380 + 400 +420 +>m * 460 ++80 + 500 i SZO +S1+0 



1^19. (k^AWt. (kvadc 
^(^(cneAlj^t. ~\W S>«Y.W«t Vest, fc, 

jo-Wo«\ VxsX.ToUesHD. XY fl »^d TED 

(oil* <\ ^ve^-< e ^.t.€.. 3*. Gv»-«.tle (v\edvi.v\. 
(Lo« t rf i^uv^) Top t^uves, ^e f to 



"?T^«.rr t *.-i;V 



j, if * /f ■3 (f - 



1W &c 



r / 7 ' t> "Wit « 



/J«W *J6 llliumu XI It 3 jj tig- 21 /j^ 'ftVi-37 •*> St i n II W ^ £&_^S'0-Y(ov<i.'\'«.sV (.G<n<ritVe.A.'N 

Figs. 2C-29. For the various lists referred to, see Appendix II. The scale for Fig. 20. is not complete, word S2 of the Rice List being at +685 from the 3d-grade median, or +1155 from absolute zero. 



Rice Sentence Test. Easy 50-Word Test 83 

to this list and to the Preferred List enable us to suggest a 
correction. Their P.E. values, when the averages of the grades 
writing them are taken, appear as follows : 

Easy 50- Preferred 

word List List Increase 

sure +.530 +1.57 1.04 

front —.299 +1.06 1.36 

touch +.652 +1.71 1.06 

only —.089 —.57 .66 

pretty —.024 +1.31 1.33 

even —.097 + .70 .80 

Average Increase 1 . 04 

It appears, therefore, that in order to compare the words of 
the Easy 50- Word List with those of the Preferred List and 
to scale them together we ought to raise all the words of the 
former list about 1 P.E. In Figure 29, accordingly, all these 
words have been raised that amount. 

Fig. 29 shows our most complete scale. It has decided limita- 
tions, and it is impossible — in the case of the newly added 
words — to suppose that it is more than an approximation. A 
great deal more testing than we have been able to do will have 
to be done before these words and others with them can be 
precisely fixed beyond dispute. It is not claimed that the scale 
we give is final. We think, however, that, supposing the two 
fundamental assumptions upon which it is based to be valid, 
it may be used in its present form with substantially accurate 
results ; and we are confident that the general method by which 
it has been derived is the one by which a final scale may ulti- 
mately be secured. 

The top figures in Fig. 29 refer to the absolute zero-point, 
taken as 470 below the 3d-grade median. It enables us to state 
not only the difference in difficulty between words but their 
relationships. We may say, for instance, that school (No. 27 
E. 50-W. L., scales at 428) is one-half as hard as grateful 
(No. 51 R. S. T., scales at 856). We may put certain facts in 
equation form as follows: 

in = y 2 light = y pigeons = % fatiguing 

is = y 2 also = y occasion = y conscientious 

the = y chicken = y approval 

and = y penny = y peculiar 



84 Spelling Ability — Its Measurement and Distribution 

Many more such statements may be made. It will, we think, 
surprise most people to learn that fatiguing is only four times 
as hard as he, or that to spell occasion shows but three times as 
much ability as to spell is. 

In fact it will, we think, be seriously questioned whether such 
words as at, of, on, do, etc., have difficulties anything like as 
great as is shown on our scale. It will be asked, What words 
can be easier than these? If a child cannot spell them does he 
not show zero ability? The answer is that if one or more of 
these very easy words were isolated and pronounced to a group 
to be written, those who could not spell them would indeed 
show no spelling ability. But these words were not isolated, 
they were given in a context. It is one thing for children to 
write the word " at " when pronounced alone or in column 
dictation. It is quite another to write it in the sentence : " They 
are not at all pretty." Some will omit it, and this fault is not 
confined by any means to the lowest classes. Some will connect 
it with the word all, because they habitually do so in speaking. 
Some little children will quite break down on the whole sentence 
because they can't get over the word " they." In other sentences 
some will substitute a word (generally of similar meaning) for 
the one dictated. Each of these faults scores " wrong," and 
none of them would be made in column dictation. It is also true 
that children writing sentences more often write illegibly than 
they do when writing a few words in columns ; and this is 
particularly true with young children. 

It will therefore be clear that the decision as to how hard a 
word is, depends on how you use it in testing and when you call 
it " wrong." To verify the placings of the words given in this 
study one ought to use the same test material and the same 
method of scoring. In particular, column dictation will not do 
at all. 

§ 19. Derived Forms of Distribution 

The foregoing treatment of the measurement of spelling 
ability has, as has been indicated frequently, proceeded upon 
the assumption that the distribution of ability is in all grades 
normal. Such an assumption has always been made in the 
investigation of school abilities by persons whose knowledge of 



Derived Forms of Distribution 85 

the theory of statistics has enabled them to do so. In Section 10 
I have said : " There seems no good ground for assuming that 
the distribution of spelling ability in any grade is not according 
to the normal curve or according to a curve which resembles it 
closely." By this alternative is suggested the possible applica- 
bility of certain curves not of normal form but resembling the 
normal form. Our problem will now be to derive and apply 
to some of our material such modifications of the type form of 
distribution as our present knowledge of grade conditions 
permits. 

In order that the frequency of measurements within a group 
may be distributed according to the Probability Integral it is 
necessary that the group be in no way selected on the basis of 
the characteristic that is measured. It must be a random 
sampling from a " total population." If the frequency distribu- 
tions of statures for adult males born in the City of New York 
may be expected to approximate the symmetrical type, the 
distributions of statures for adult males on the police force of 
New York City would not do so. Their curve will be of the 
" moderately asymmetrical type " being cut off at the low end 
because extremely short men are at a disadvantage in the group 
supposed to be measured. In other words, there is a selection 
on the basis of stature. The group " adult males on the police 
force of New York City " is not a random sampling of the total 
population " adult males of New York City." 

The question then is: To what extent does the membership 
of each grade of the elementary school fail of being a chance 
selection from a total population ? We may fairly assume, in the 
first place, that the pupils of the first and second grades are 
unselected. Practically all children attend school and none drop 
out in these grades. From the 3d grade on, however, each 
successive grade constitutes a group which is less and less a 
random sampling. Many influences are at work to eliminate 
a greater and greater number of individuals. Probably the most 
important of them is the inability of children to progress — i.e., 
lack of ability in the lines of work now required by the schools. 

The extent to which elimination takes place in the grades has 
been the subject of study by a number of investigators. The 
first of these was Thorndike ('07). He draws conclusions from 



86 Spelling Ability — Its Measurement and Distribution 

conditions in 23 cities as they were about 1900. He estimates 
that out of 100 entering pupils, 97 remain till grade 3, 90 till 
grade 4, 81 till grade 5, 68 till grade 6, 54 till grade 7, and 40 
till the last grammar grade (8th or 9th). Ayres ('09) sharply 
criticised these figures, stating that they were too small. He 
contended, particularly, that there was no dropping out before 
the 6th year — a conclusion which common observation and later 
investigation unite to disprove. Employment certificates are 
granted in great numbers to 5th-grade children. Mr. Ayres' 
figures for retention are as follows: Grades 1-5, 100 (i.e., no 
elimination) ; grade 6, 90; grade 7, 71 ; grade 8, 51. 

Thorndike, using later and better reports, subsequently derived 
figures a little higher than his former ones, but substantially 
in agreement with them (Thorndike, '10). They were no higher 
probably than 5 or 6 years of agitation would have led one to 
expect. 

Another important study of this question was made by 
Strayer ('11), the material being used from 318 cities. His 
conclusions tend to group with Thorndike's rather than with 
those of Ayres. Owing to 'the large number of cities whose 
returns were used, the uniform method of taking the census, 
and the recency of the conditions studied, this investigation is 
highly important. No single figures are given for retention in 
general, though they are easily found. Using the largest age 
group as the number of entering children, he gives the following 
as *the median per cents in each grade. 





Cities of Over 25,000 


Cities of Less than 25,000 




Boys 


Girls 


Boys 


Girls 


3d year 


115 

110 

100 

85 

65 

50 


110 
110 
95 
85 
75 
60 


110 
105 
95 
80 
70 
50 


105 


4th " 


100 


5th " 


95 


6th " 


85 


7th " 


70 


8th " 


60 







Since in this study we group boys and girls together and consider 
general conditions, the average of these percentages will give 
figures for retention for each grade (subject to deduction for 



Derived Forms of Distribution 



87 



repeaters) as follows: 3d grade, no; 4th grade, 106; 5th grade, 
96; 6th grade, 84; 7th grade, 70; 8th grade, 55. If, as Dr. 
Strayer says, a fair estimate of the number of repeaters in the 
6th, 7th, and 8th grades would be 12%, 10%, and 8% of the 
pupils in each grade (p. 136), it is likely that the progression 
(8, 10, 12) may be carried back to the 5th, 4th and 3d grades 
without great violence to the facts. We estimate therefore that 
the number of repeaters in the 3d, 4th, and 5th grades is 18%, 
16%, and 14% of the pupils in each grade. Making these de- 
ductions from the above percentages, we have for the retention : 
3d grade, 92 ; 4th grade, 90 ; 5th grade, 82 ; 6th grade, 72 ; 7th 
grade, 60; 8th grade, 47. 

Weighing as best we can the results of these four studies, 
we have made the best estimate we can for the probable amount 
of retention at present in the grades. For reasons that will 
appear later we have expressed this estimate in numbers per 
10,000 instead of per 100. 

Table XXXIII and Fig. 30 show the percentages we have 
adopted compared with those of Thorndike, Ayres, and Strayer 
(as derived). Fig. 30 gives only the earlier of Thorndike's 
percentages. 

TABLE XXXIII 
Percentages of Retention. Grades 3 to 8 



8th 



Thorndike '07 

Ayres '09 

Thorndike '10 

Strayer '11 (derived) . 

Adopted 



3d 


4th 


5th 


6th 


7th 


97 


90 


81 


68 


54 


100 


100 


100 


90 


71 




91 


81.5 


70.9 


56 


92 


90 


82 


72 


60 


97.25 


95.46 


88.40 


70.87 


57.44 



40 
51 

41.2 
47 



48.21 



Such an amount of retention for each grade having been 
adopted, the next question to consider is : W T hat part of a 
normal distribution is thus eliminated? Obviously not all the 
poorest in ability drop out. Our results for spelling show that 
some very poor spellers are retained even in the highest grades. 
Yet the greatest elimination will no doubt be among those of 
lowest ability and will be progressively less among children of 
greater ability. How much this amounts to for successive incre- 



88 Spelling Ability — Its Measurement and Distribution 

ments of ability we do not positively know. We are again forced 
to make as reasonable an estimate as we can, and this time 
without the help of any investigations. 



100 
So 

b'O 

40 



30 



Tig.30 







, — Cl^opTed. 



Fig. 30. The horizontal scale is for the grades of the elementary school ; 
the vertical scale is for percentage of pupils retained. 

We have estimated (Table XXXIII) that 9725 out of 10,000 en- 
tering children are retained in the 3d grade ; and we judge further 
that all below — 4 P.E. have dropped out, that at — 3 P.E. 40% 
have dropped out, and that at — 2 P.E. none at all. In the 4th 
grade, retention is put at 9546 in 10,000 (Table XXXIII) and 
total elimination is estimated to operate up to — 3.7 P.E. From this 
point to — 2.y P.E. the force of elimination is supposed gradually 
to diminish until 40% drop out. At — 1.7 P.E. only 10% are 
estimated as lost to the grade. At — 1.2 P.E. the forces tending 
to eliminate children are supposed to have been completely 
counteracted by the opposing forces tending to retain them. In 
the 5th grade (8840 out of 10,000 retained) the elimination is 



Derived Forms of Distribution 



89 



assumed to be total up to — 3.2 P.E. and partial as high as 
— 0.2 P.E. In the 6th grade (7087 out of 10,000 retained) the 
corresponding points are — 2.J P.E. and + 3.3 P.E. ; in the 7th 
(5744 retained) they are — 2.3 P.E. and + 3.7 P.E. ; and in the 
8th (4821 retained), — 2 P.E. and + 6 P.E. It is, therefore, sup- 
posed that in the last three years of the elementary school there 
is some elimination even among the most capable children. This 
plan of elimination and retention may easily be attacked as 
artificial and it may very likely be shown to need considerable 
modification when later and better knowledge is available on this 
difficult subject. Meanwhile, however, some assumption was 
necessary in order to construct any sort of frequency tables 
which should illustrate the method of constructing a scale when 
account is taken of the selective influence of the grades. We can 
only state that we have keenly appreciated the importance of 
distributing the amount of elimination where it most probably 
occurs, that we have been at no small pains to find out where 
to distribute it, and that we have estimated as wisely as we could. 
Table XXXIV gives the entire plan of elimination and retention 

for each grade. 

s TABLE XXXIV 

Plan of Elimination and Retention for Each Grade 





[X 


Per 


Per 




X 


Per 


Per 


Grade, etc. 




cent 


cent 


Grade, etc. 





cent 


cent 




P.E. 


eliminated 


retained 




P.E. 


eliminated 


retained 


3d Grade. 


—4 


100 





7th Grade 


—2.3 


100 





#=9725 . 


—3 


40 


60 


#=5744 . 


—1.3 


70 


30 




—2 





100 




—0.3 

+0.7 
+ 1.7 


50 
20 
10 


50 

80 
90 


4th Grade 


—3.7 


100 







#=9546 . 


—2.7 


40 


60 




+2.7 


5 


95 




—1.7 


10 


90 




+3.7 





100 




—1.2 





100 












8th Grade 
#=4821 . 


—2 
—1 


100 
70 




30 


5th Grade 


—3.2 


100 





#=8840 . 


—2.2 


50 


50 







50 


50 




—1.2 


20 


80 




+ 1 


30 


70 




—0.2 





100 




+2 
+3 
+4 


20 

10 

6 


80 
90 
94 


6th Grade 


—2.7 


100 







#=7087 . 


—1.7 


60 


40 




+ 5 


2 


98 




—0.7 


35 


65 




+ 6 





100 




+0.3 


15 


85 












+ 1.3 


10 


90 












+2.3 


5 


95 












+ 3.3 





100 











90 Spelling Ability — Its Measurement and Distribution 







-JfPL. ~3te -ZPt. -"?«• +P& **?*■ +-3/?d +¥?■& 



^U. -3P.&. -2P.£ -?£ 



^Gcvccde- 




-fVS -tZ?l +3 Pi. **Pd. 



Figs. 31-36. The estimated amount and distribution of elimination and 
retention. See Table XXXIII. 



Derived Forms of Distribution g i 

The next step was to apply the data of Table XXXIV to the 
normal distribution and to derive therefrom for each grade a 
modified distribution which should take account of the amount 
and range of elimination as estimated. In order that the validity 
of our method may be open to inspection, we shall illustrate for 
the 6th grade the manner in which these modified distributions 
were derived. 

We have adopted certain percentages of retention for desig- 
nated amounts of general ability (Table XXXIV, 6th grade), 
and these percentages must not only stand the test of reasonable- 
ness in themselves, but they must also when applied to a normal 
table of frequency (the sum of whose cases is, say, iooo), reduce 
the number of cases to an amount which represents a reasonable 
percentage of retention for the 6th grade (say, 70 or 71). That 
is, the derived table must show approximately 700 cases out of 
1000, or 7000 out of 10,000. We shall see later to what extent 
this turns out to be true. 

Adapting the normal table of frequency (Table XIV, page 35) 
so as to include 1000 cases instead of 10,000 and taking intervals 
of 0.1 P.E., we have columns 1 and 2 of Table XXXV. In 
column 3 we increase the percentages of retention from o at 

— 2.7 P.E. to 40 at — 1.7 P.E. by increments of 4 for each of the 
ten steps; then by increments of 2.5 until 65 is reached at 

— 0.7 P.E. ; then by increments of 2 to 85 at -f 0.3 P.E. ; and. 
so on as required by Table XXXIV, col. 4. Taking these 
percentages of the frequencies in column 2 gives the derived 
frequencies of column 4. The sum of the entries in this column 
being 708.7, the plan gives an amount of elimination which is 
reasonable for the 6th grade. (See Table XXXIII.) 

The amount and distribution of elimination and retention are 
shown by diagram for each grade in Figs. 31 to 36. Fig. 34 
in particular shows these facts for the 6th grade, and is the 
graphic representation of the series of frequencies in column 4 
of Table XXXV. Fig. 31 shows the same facts for the 3d 
grade, Fig. ^2 for the 4th grade, etc. The progressive increase 
in elimination and the extension of it to higher and higher parts 
of the normal curve are the facts to be noticed. 

But we have not in column 4 of Table XXXV, a frequency 
table for the 6th grade in the most useful form. The area of its 



92 Spelling Ability — Its Measurement and Distribution 



TABLE XXXV 

Sixth Grade. Derivation of Modified Table of Frequency 

Below Normal Median Above Normal Median 







Per- 










Per- 








Normal 


cent- 


Derived 


Same on 




Normal 


cent 


Derived 


Same on 


X 


Fre- 


ages 


Fre- 


basis of 


X 


Fre- 


ages 


Fre- 


basis of 





quen- 


of 


quen- 


10,000 





quen- 


of 


quen- 


10,000 


P.E. 


cies 


Reten- 
tion 


cies 


cases 


P.E. 


cies 


Reten- 
tion 


cies 


cases 


0— .1 


27 


79 


21.3 


301 


0— .1 


27 


81 


21.9 


309 


.2 


27 


77 


20.8 


293 


.2 


27 


83 


22.4 


312 


.3 


26 


75 


19.5 


275 


.3 


26 


85 


22.1 


316 


.4 


26 


73 


19.0 


268 


.4 


26 


85.5 


22.2 


316 


.5 


26 


71 


18.5 


261 


.5 


26 


86 


22.4 


313 


.6 


25 


69 


17.3 


244 


.6 


25 


86.5 


21.6 


308 


.7 


25 


67 


16.8 


237 


.7 


25 


87 


21.8 


305 


.8 


23 


65 


15.0 


212 


.8 


23 


87.5 


20.1 


284 


.9 


23 


62.5 


14.4 


203 


.9 


23 


88 


20.2 


285 


1.0 


22 


60 


13.2 


186 


1.0 


22 


88.5 


19.5 


275 


1.1 


21 


57.5 


12.1 


171 


1.1 


21 


89 


18.7 


264 


1.2 


20 


55 


11.0 


155 


1.2 


20 


89.5 


17.9 


252 


1.3 


19 


52.5 


10.0 


141 


1.3 


19 


90 


17.1 


241 


1.4 


18 


50 


9.0 


127 


1.4 


18 


90.5 


16.3 


230 


1.5 


16 


47.5 


7.6 


107 


1.5 


16 


91 


14.6 


206 


1.6 


16 


45 


7.2 


102 


1.6 


16 


91.5 


14.6 


206 


1.7 


14 


42.5 


6.0 


85. 


1.7 


14 


92 


12.9 


182 


1.8 


14 


40 


5.6 


79 


1.8 


14 


92.5 


13.0 


183 


1.9 


12 


36 


4.3 


61 


1.9 


12 


93 


11.2 


158 


2.0 


11 


32 


3.5 


50 


2.0 


11 


93.5 


10.3 


145 


2.1 


11 


28 


3.1 


44 


2.1 


11 


94 


10.3 


145 


2.2 


9 


24 


2.2 


31 


2.2 


9 


94.5 


8.5 


121 


2.3 


9 


20 


1.8 


26 


2.3 


9 


95 


8.6 


120 


2.4 


7 


16 


1.1 


16 


2.4 


7 


95.5 


6.7 


95 


2.5 


7 


12 


.8 


11 


2.5 


7 


96 


6.7 


95 


2.6 


6 


8 


.5 


7 


2.6 


6 


96.5 


5.8 


82 


2.7 


6 


4 


.2 


3 


2.7 


6 


97 


5.8 


82 


2.8 


5 











2.8 


5 


97.5 


4.9 


69 












2.9 


4 


98 


3.9 


55 












3.0 


4 


98.5 


3.9 


55 












3.1 


3 


99 


3.0 


42 












3.2 


3 


99.5 


3.0 


42 












3.3 


2 


100 


2.0 


28 












3.4 


2 


100 


2.0 


28 












3.5 


2 


100 


2.0 


28 












3.6 


2 


100 


2.0 


28 












3.7 


1 


100 


1.0 


14 












3.8 


1 


100 


1.0 


14 












3.9 


1 


100 


1.0 


14 












4.0 


1 


100 


1.0 


14 












etc. to 


etc. to 


etc. 


etc. to 


etc. to 




tal No. 


cases. . 






6.0 


.02 




.02 


.28 


Tc 




1000 




708.7 


9999.72 













Derived Forms of Distribution 



93 



curve is no longer iooo, but only 708.7. In order to express the 
several frequencies in the form of per cents, we shall have to 
divide each of them (column 4) by their total (708.7). Express- 
ing these quotients on the basis of 10,000 instead of 1000, we have 
column 5. These are the numbers in the columns 3 and 5 of 
Table XXXIX (p. 96) ; and when their sums are taken begin- 
ning at o they constitute the Modified Table of Frequency for 
the 6th grade (Table XXXIX). 



TABLE XXXVI 

Modified Table of Frequency, 3d Grade. Median =4-0.051 P.E. 
Plan of elimination: —4 P.E., 100%; —3 P.E., 40%; —2 P.E., 0% 
Total area of the surface of frequency taken as 10,000. See Fig. 37. 



X 


Low 


High 


X 


Low 


High 


X 

P.E. 


Low 


High 


P.E. 


% 


A 


% 


A 


P.E. 


% 


A 


% 


A 


% 


A 


% 


A 






278 




278 




109 




113 










5.1 


.1 


278 


278 


278 


278 


2.1 


4334 


85 


4338 


93 


4.1 






5116.1 


5.1 


.2 


556 


267 


556 


267 


2.2 


4419 


81 


4431 


93 


4.2 






5121.2 


5.1 


.3 


823 


267 


823 


267 


2.3 


4500 


60 


4524 


72 


4.3 






5126.3 


4.1 


.4 


1090 


267 


1090 


267 


2.4 


4560 


58 


4596 


72 


4.4 






5130.4 


2.1 


.5 


1357 


257 


1357 


257 


2.5 


4618 


47 


4668 


62 


4.5 






5132.5 


2.1 


.6 


1614 


257 


1614 


257 


2.6 


4665 


44 


4730 


62 


4.6 






5134.6 


1.0 


.7 


1871 


236 


1871 


236 


2.7 


4709 


35 


4792 


51 


4.7 






5135.6 


1.0 


.8 


2107 


236 


2107 


236 


2.8 


4744 


26 


4843 


41 


4.8 






5136.6 


1.0 


.9 


2343 


226 


2343 


226 


2.9 


4770 


25 


4884 


41 


4.9 






5137.6 


1.0 • 


1.0 


2569 


216 


2569 


216 


3.0 


4795 


17 


4925 


31 


5.0 






5138.6 


.51 


1.1 


2785 


206 


2785 


206 


3.1 


4812 


15 


4956 


31 


5.1 






5139.11 


.51 


1.2 


2991 


195 


2991 


195 


3.2 


4827 


9 


4987 


21 


5.2 






5139.62 


.31 


1.3 


3186 


185 


3186 


185 


3.3 


4836 


7.4 


5008 


21 


5.3 






5139.93 


.31 


1.4 


3371 


165 


3371 


165 


3.4 


4843.4 


6.2 


5029 


21 


5.4 






5140.24 


.31 


1.5 


3536 


165 


3536 


165 


3.5 


4849.6 


4.9 


5050 


21 


5.5 






5140.55 


.31 


1.6 


3701 


144 


3701 


144 


3.6 


4854.5 


1.9 


5071 


10 


5.6 






5140.86 


.21 


1.7 


3845 


144 


3845 


144 


3.7 


4856.4 


1.2 


5081 


10 


5.7 






5141.07 


.21 


1.8 


3989 


123 


3989 


123 


3.8 


4857.6 


0.6 


5091 


10 


5.8 






5141.28 


.21 


1.9 


4112 


113 


4112 


113 


3.9 


4858.2 




5101 


10 


5.9 






5141.49 


.21 


2.0 


4225 




4225 




4.0 






5111 




6.0 






5141.70 





94 Spelling Ability — Its Measurement and Distribution 



TABLE XXXVII 
Modified Table of Frequency, 4th Grade. Median= +0.087 P.E. 
Plan of elimination: —3.7 P.E., 100%; —2.7 P.E., 40%; —1.7 P.E., 
10%; — 1.2 P.E., 0%. Total area of the surface of frequency taken as 
10,000. See Fig. 38. 



X 


Low 


High 


X 


Low 


High 


X 


Low 


High 


P.E. 


% 


A 


% 


A 


P.E. 


% 


A 


% 


A 


P.E. 


% 


A 


% 


A 






283 




283 






93 




115 










5.2 


.1 


283 


283 


283 


283 


2.1 


4316 


74 


4421 


94 


4.1 






5210.2 


5.2 


.2 


566 


272 


566 


272 


2.2 


4390 


71 


4515 


94 


4.2 






5215.4 


5.2 


.3 


838 


272 


838 


272 


2.3 


4461 


53 


4609 


73 


4.3 






5220.6 


4.2 


.4 


1110 


271 


1110 


271 


2.4 


4514 


51 


4682 


73 


4.4 






5224.8 


2.1 


.5 


1381 


262 


1381 


262 


2.5 


4565 


41 


4755 


63 


4.5 






5226.9 


2.1 


.6 


1643 


262 


1643 


262 


2.6 


4606 


40 


4818 


63 


4.6 






5229 


1.05 


.7 


1905 


241 


1905 


241 


2.7 


4646 


31 


4881 


52 


4.7 






5230.05 


1.05 


.8 


2146 


241 


2146 


241 


2.8 


4677 


23 


4933 


42 


4.8 






5231.10 


1.05 


.9 


2387 


230 


2387 


230 


2.9 


4700 


20 


4975 


42 


4.9 






5232.15 


1.05 


1.0 


2617 


220 


2617 


220 


3.0 


4720 


13 


5017 


31 


5.0 






5233.20 


.52 


1.1 


2837 


210 


2837 


210 


3.1 


4733 


11 


5048 


31 


5.1 






5233.72 


.52 


1.2 


3047 


199 


3047 


199 


3.2 


4744 


6 


5079 


21 


5.2 






5234.24 


.31 


1.3 


3246 


185 


3246 


189 


3.3 


4750 


5 


5100 


21 


5.3 






5234.55 


.31 


1.4 


3431 


161 


3435 


168 


3.4 


4755 


4 


5121 


21 


5.4 






5234.86 


.31 


1.5 


3592 


158 


3603 


168 


3.5 


4759 


3 


5142 


21 


5.5 






5235.17 


.31 


1.6 


3750 


135 


3771 


147 


3.6 


4762 


1 


5163 


10.5 


5.6 






5235.48 


.21 


1.7 


3885 


132 


3918 


147 


3.7 


4763 




5173.5 


10.5 


5.7 






5235.69 


.21 


1.8 


4017 


109 


4065 


126 


3.8 






5184 


10.5 


5.8 






5235.90 


.21 


1.9 


4126 


97 


4191 


115 


3.9 






5194.5 


10.5 


5.9 






5236.11 


.21 


2.0 


4223 




4306 




4.0 






5205 




6.0 






5236.32 





Derived Forms of Distribution 



95 



TABLE XXXVIII 
Modified Table of Frequency, 5th Grade. Median= + 0.215 P.E. 
Plan of elimination: —3.2 P.E., 100%; —2.2 P.E., 50%; —1.2 P. E., 
20%; — 0.2 P.E., 0%. Total area of the surface of frequency taken as 
10,000. See Fig. 39. 



X 


Low 


High 


X 


Low 


High 


X 


Low 


High 


P.E. 


% 


A 


% 


A 
305 


P.E. 


% 


A 


% 


A 


P.E. 


% 


A 


% 


A 






305 








70 




124 










6 


.1 


305 


305 


305 


305 


2.1 


4093 


54 


4772 


102 


4.1 






5627 


6 


.2 


610 


294 


610 


294 


2.2 


4147 


51 


4874 


102 


4.2 






5633 


6 


.3 


904 


288 


904 


294 


2.3 


4198 


36 


4976 


79 


4.3 






5639 


5 


.4 


1192 


282 


1198 


294 


2.4 


4234 


32 


5055 


79 


4.4 






5644 


2 


.5 


1474 


266 


1492 


283 


2.5 


4266 


24 


5134 


68 


4.5 






5646 


2 


.6 


1740 


260 


1775 


283 


2.6 


4290 


20 


5202 


68 


4.6 






5648 


1 


.7 


2000 


234 


2058 


260 


2.7 


4310 


14 


5270 


57 


4.7 






5649 


1 


.8 


2234 


229 


2318 


260 


2.8 


4324 


9 


5327 


45 


4.8 






5650 


1 


.9 


2463 


214 


2578 


249 


2.9 


4333 


7 


5372 


45 


4.9 






5651 


1 


1.0 


2677 


200 


2827 


238 


3.0 


4340 


3 


5417 


34 


5.0 






5652 


.6 


1.1 


2877 


186 


3065 


226 


3.1 


4343 


2 


5451 


34 


5.1 






5652.6 


.5 


1.2 


3063 


172 


3291 


215 


3.2 


4345 




5485 


23 


5.2 






5653.1 


.3 


1.3 


3235 


157 


3506 


204 


3.3 






5508 


23 


5.3 






5653.4 


.3 


1.4 


3392 


134 


3710 


181 


3.4 






5531 


23 


5.4 






5653.7 


.3 


L.5 


3526 


129 


3891 


181 


3.5 






5534 


23 


5.5 






5654.0 


.3 


1.6 


3655 


108 


4072 


158 


3.6 






5577 


11 


5.6 






5654.3 


.2 


1.7 


3763 


103 


4230 


158 


3.7 






5588 


11 


5.7 






5654.5 


.2 


1.8 


3866 


84 


4388 


136 


3.8 






5599 


11 


5.8 






5654.7 


.2 


1.9 


3950 


73 


4524 


124 


3.9 






5610 


11 


5.9 






5654.9 


.2 


2.0 


4023 




4648 




4.0 






5621 




6.0 






5655.1 





96 Spelling Ability — Its Measurement and Distribution 



TABLE XXXIX 

Modified Table of Frequency, 6th Grade. Median =+0.418P.E. 

Plan of elimination: —2.7 P.E., 100%; —1.7 P.E., 60%; —0.7 P.E., 
35%; +0.3 P.E., 15%; +1.3 P.E., 10%; +2.3 P.E., 5%; +3.3 P.E., 
0%. Total area of surface of frequency taken as 10,000. See Fig. 40. 



X 


Low 


High 


X 

P.E. 


Low 


High 


X 

P.E. 


Low 


High 


P.E. 


% 


A 


% 


A 


% 


A 


% 


A 


% 


A 


% 


A 






301 




309 






44 




145 










7 


.1 


301 


293 


309 


312 


2.1 


3602 


31 


5235 


121 


4.1 






6268 


7 


.2 


594 


275 


621 


316 


2.2 


3633 


26 


5356 


120 


4.2 






6275 


7 


.3 


869 


26S 


937 


316 


2.3 


3659 


16 


5476 


95 


4.3 






8282 


6 


.4 


1137 


261 


1253 


313 


2.4 


3675 


11 


5571 


95 


4.4 






6288 


3 


.5 


1398 


244 


1566 


308 


2.5 


3686 


7 


5666 


82 


4.5 






6291 


3 


.6 


1642 


237 


1874 


305 


2.6 


3693 


3 


5748 


82 


4.6 






6294 


1.4 


.7 


1879 


212 


2179 


284 


2.7 


3696 




5830 


69 


4.7 






6295.4 


1.4 


.8 


2091 


203 


2463 


285 


2.8 






5899 


55 


4.8 






62968 


1.4 


.9 


2294 


186 


2748 


275 


2.9 






5954 


55 


4.9 






6298.2 


1.4 


1.0 


2480 


171 


3023 


264 


3.0 






6009 


42 


5.0 






6299.6 


.7 


1.1 


2651 


155 


3287 


252 


3.1 






6051 


42 


5.1 






6300.3 


.7 


1.2 


2806 


141 


3539 


241 


3.2 






6093 


28 


5.2 






6301 


.4 


1.3 


2947 


127 


3780 


230 


3.3 






6121 


28 


5.3 






6301.4 


.4 


1.4 


3074 


107 


4010 


206 


3.4 






6149 


28 


5.4 






6301.8 


.4 


1.5 


3181 


102 


4216 


206 


3.5 






6177 


28 


5.5 






6302.2 


.4 


1.6 


3283 


85 


4422 


182 


3.6 






6205 


14 


5.6 






6302.6 


.28 


1.7 


3368 


79 


4604 


183 


3.7 






6219 


14 


5.7 






6302.88 


.28 


1.8 


3447 


61 


4787 


158 


3.8 






6233 


14 


5.8 






6303.16 


.23 


1.9 


3508 


50 


4945 


145 


3.9 






6247 


14 


5.9 






6303.44 


.28 


2.0 


3558 




5090 




4.0 






6261 




6.0 






6303.72 





Derived Forms of Distribution 



97 



TABLE XL 

Modified Table of Frequency, 7th Grade. Median= +0.669 P.E. 

Plan of elimination: —2.3 P.E., 100%; —1.3 P.E., 70%; —0.3 P.E., 
50%) ; +0.7 P.E., 20%; +1.7 P.E., 10%; +2.7 P.E., 5%; +3.7 P.E., 
0%. Total area of frequency surface taken as 10,000. See Fig. 41. 



X 


Low 


High 


X 


Low 


High 


X 

P.E. 


Low 


High 


P.E. 


% 


A 


% 


A 


P.E. 


% 


A 


% 


A 


% 


A 


% 


A 






277 




290 






17 




176 










9 


.1 


277 


263 


290 


306 


2.1 


2844 


9 


5846 


145 


4.1 






7096 


9 


.2 


540 


240 


596 


308 


2.2 


2853 


5 


5991 


145 


4.2 






7105 


9 


.3 


780 


226 


904 


322 


2.3 


2858 




6136 


114 


4.3 






7114 


7 


.4 


1006 


217 


1226 


334 


2.4 






6250 


114 


4.4 






7121 


3 


.5 


1223 


200 


1560 


336 


2.5 






6364 


99 


4.5 






7124 


3 


.6 


1423 


191 


1896 


348 


2.6 






6463 


99 


4.6 






7127 


2 


.7 


1614 


169 


2244 


326 


2.7 






6562 


83 


4.7 






7129 


2 


.8 


1783 


160 


2570 


326 


2.8 






6645 


66 


4.8 






7131 


2 


.9 


1943 


146 


2896 


318 


2.9 






6711 


68 


4.9 






7133 


2 


1.0 


2089 


132 


3214 


306 


3.0 






6779 


50 


5.0 






7135 


.9 


1.1 


2221 


118 


3520 


296 


3.1 






6829 


50 


5.1 






7135.9 


.9 


1.2 


2339 


106 


3816 


283 


3.2 






6879 


35 


5.2 






7136.8 


.5 


1.3 


2445 


94 


4099 


273 


3.3 






6914 


35 


5.3 






7137.3 


.5 


1.4 


2539 


75 


4372 


246 


3.4 






6949 


35 


5.4 






7137.8 


.5 


1.5 


2614 


66 


4618 


246 


3.5 






6984 


35 


5.5 






7138.3 


.5 


1.6 


2680 


50 


4864 


220 


3.6 






7019 


17 


5.6 






7138.8 


.35 


1.7 


2730 


43 


5084 


220 


3.7 






7036 


17 


5.7 






7139.15 


.35 


1.8 


2773 


31 


5304 


190 


3.8 






7053 


17 


5.8 






7139.50 


.35 


1.9 


2804 


23 


5494 


176 


3.9 






7070 


17 


5.9 






7139.85 


.3 


2.0 


2827 




5670 




4.0 






7087 




6.0 






7140.15 





98 Spelling Ability — Its Measurement and Distribution 



TABLE XLI 

Modified Table of Frequency, 8th Grade. Median= +0.746 P.E. 

Plan of elimination: —2 P.E., 100%; —1 P.E., 70%; P.E., 50%; +1 
P.E., 30%; +2P.E., 20%; +3 P.E., 10%; + 4 P.E., 6%; +5 P.E., 2%; 
+ 6 P.E., 0%. Total area of frequency surface taken as 10,000. See Fig. 42. 



X 


Low 


High 


X 

P.E. 


Low 


High 


X 

P.E. 


Low 


High 


P.E. 


% 


A 


'% 


A 


% 


A 


% 


A 


% 


A 


% 


A 






280 




290 










184 










10 


.1 


280 


270 


290 


303 


2.1 






5834 


165 


4.1 






7195 


10 


.2 


550 


249 


593 


303 


2.2 






5999 


145 


4.2 






7205 


10 


.3 


799 


237 


896 


313 


2.3 






6144 


130 


4.3 






7215 


8 


.4 


1036 


226 


120? 


323 


2.4 






6274 


116 


4.4 






7223 


4 


.5 


1262 


207 


1532 


323 


2.5 






6390 


108 


4.5 






7227 


4 


.6 


1469 


197 


1855 


332 


2.6 






6498 


108 


4.6 






7231 


2 


.7 


1666 


172 


2187 


320 


2.7 






6606 


91 


4.7 






7233 


2 


.8 


1838 


162 


2507 


319 


2.8 






6697 


75 


4.8 






7235 


2 


.9 


2000 


145 


2826 


319 


2.9 






6772 


75 


4.9 






7237 


2 


1.0 


2145 


131 


3145 


309 


3.0 






6847 


56 


5.0 






7239 


1 


1.1 


2276 


112 


3454 


299 


3.1 






6903 


56 


5.1 






7240 


1 


1.2 


2388 


95 


3753 


288 


3.2 






6959 


38 


5.2 






7241 


.6 


1.3 


2483 


79 


4041 


276 


3.3 






6997 


38 


5.3 






7241.6 


.6 


1.4 


2562 


60 


4317 


260 


3.4 






7035 


37 


5.4 






7242.2 


.6 


1.5 


2622 


50 


4577 


242 


3.5 






7072 


37 


5.5 






7242.8 


.6 


1.6 


2672 


35 


4819 


235 


3.6 






7109 


19 


5.6 






7243.4 


.4 


1.7 


2707 


27 


5054 


215 


3.7 






7128 


19 


5.7 






7243.8 


.4 


1.8 


2734 


14 


5269 


197 


3.8 






7147 


19 


5.8 






7244.2 


.4 


1.9 


2748 


6 


5466 


184 


3.9 






7166 


19 


5.9 






7244.6 


.4 


2.0 


2754 




5650 




4.0 






7185 




6.0 






7245 





In this manner each of the Modified Tables of Frequency was 
made up. They are given in tables XXXVI to XLI. They 
are intended to take the place, each for the grade to which it 
applies, of the Table of Frequency for the normal distribution. 
Since they are asymmetrical, the lower and upper parts have to 
be given separately. For the same reason there is no P.E., the 
use of the Probable Error as a unit of amount being properly 
confined to normal curves only (Yule, 'n, p. 147). The 
quartile deviation (Q3 — Qi) might be used instead, but its 



Derived Forms of Distribution 



99 








kJH 




/A 








c 


> c 


> 





Figs. 37-42. Derived Forms of Distribution. Grades 3 to 8. 



ioo Spelling Ability — Its Measurement and Distribution 

value differs for each of the six tables. In order therefore to 
employ a unit which should be the same for all, including the 
normal distribution, we have retained the P.E. of the Proba- 
bility Integral. It is now no longer a function of the modified dis- 
tributions, but a mere unit of length. Likewise in order to 
have a common point of reference the median of the normal 
distribution has been retained, the terms " low " and " high " in 
the tables referring to parts below or above that point. The 
real median of each modified distribution, however, is given, 
being expressed as a deviation from the old median. 

Figs. 37 to 42 are to be considered in connection with tables 
XXXVI to XLI, of which they are the graphic expressions (the 
curves being " smoothed/' to represent an indefinite number of 
cases) . They are also to be considered in connection with Figs. 31 
to 36, from the " retention " parts of which they are derived 
by making the areas 10,000. In Figs. 37 to 42, the curve extend- 
ing farther to the left is in each case the normal curve and 
OM is its median vertical. 1 M 1 is the median vertical of the 
modified surface and OO 1 is the distance between medians. The 
values of these are as follows: 3d grade, 0.051 P.E. ; 4th 
grade, 0.087 P-E. ; 5th grade, 0.215 P.E. ; 6th grade, 0.418 P.E. ; 
7th grade, 0.669 P-E- ; 8th grade, 0.746 P.E. 

Is it worth while to use these tables instead of the normal 
one? Will the same material when analyzed by the skew and 
normal distributions yield differences that are important? With 
the purpose of throwing some light on this question we have 
used the modified tables to interpret the results of testing with our 
Preferred List, and the rest of the present section will be 
devoted to this matter. The differences will not In many cases 
be found to be large. This is, of course, particularly true when 
the early grades are concerned, the curves being for those grades 
almost normal. It may be remarked, however, that the applica- 
bility of these tables does not rest upon the results here shown. 
It is general ability rather than spelling ability that tends 
strongly to keep children in school. Spelling ability does not 
correlate as highly with general ability as do the abilities 
in most other school subjects. It is quite likely, therefore, that 



Derived Forms of Distribution 



101 



the use of these tables for the statistical treatment of other 
subjects may be more satisfactory than it is for spelling. They 
are given here primarily to illustrate the method. 

TABLE XLII 
Number and Per Cent of Pupils in Each Grade Whose Ability 
Equalled or Exceeded that of the Median Pupil in Every Other Grade 
with the P.E. Values Corresponding to Each Per Cent. Selected 
List. Modified Distributions. Compare with Table XV (p. 36) 







3d 

Grade 


4th 

Grade 


5th 
Grade 


6th 

Grade 


7th 
Grade 


8th 
Grade 


3d grade. . . 
#=445 


No. 

% 
P.E. 




76 
17.1 
1.3858 


27 
6.1 

2.2736 


9 
2.0 
3.0029 


3 
0.7 

3.607 




? 


4th grade . . 
#=467 


No. 

% 
P.E. 


378 

80.9 

1.1949 




146 
31.3 
.6965 


52 
11.1 
1.7616 


27 

5.8 

2.2778 


9 
1.9 

3.0076 


5th grade . . 
#=515 


No. 

% 
P.E. 


478 

92.8 

1.7916 


370 
71.8 
.7341 




142 

27.6 
.8136 


73 
14.2 
1.5172 


30 
5.8 
2.2104 


6th grade . . 
#=418 


No. 

% 
P.E. 


414 

99.0 
2.5043 


384 

91.9 

1.6747 


338 

80.1 

1.0451 




142 

34.0 
.5386 


57 
13.6 
1.4812 


7th grade . . 
#=365 


No. 

% 
P.E. 


363 

99.5 

2.6038 


354 

96.4 

2.0360 


328 

89.9 

1.5096 


256 

70.1 

.6586 




99 
27.1 
.7564 


8th grade . . 

#=277 


No. 

% 
P.E. 


227 
100 
? 


276 

99.6 

2.4719 


269 

97.1 

2.0197 


241 

87.0 
1.4424 


200 
72.2 
.635 





In Section n we located the grade medians assuming normal 
distribution. In Tables XLII and XLIII the same data have 
been subjected to analysis using the modified distributions. 
These tables are to be compared with Tables XV (page 36) and 
XVI (page 39). The median intervals are considerably less 
than they were found to be by using the normal distribution. 
Note the comparisons in Table XLIV (page 103). 

On the average, the intervals by the present method are less 
than the same intervals found by using the normal distribution 
by 0.1247 P.E., or about half a step in the 10-point scale (Table 
XX, page 52). Since this occurs five times the entire range 



io2 Spelling Ability — Its Measurement and Distribution 



TABLE XLIII 

Direct and Derived Values of Median Distances. Modified Distribu- 
tions. Selected List 





^3- 4 


^4-5 


^5-6 


^6-7 


^7-8 




1.3858 
(direct) 


.8878 
(M 3 _ 5 ~M 3 _ A ) 


.7293 
(M 3 _ Q — M 3 _ h ) 


.6041 
(M 3 _ 7 — M 3 _ 6 ) 


? 

(M 3 _ 8 — M 3 _ 7 ) 




1.5771 
(M 3 _ 5 ~M 4 _ 5 ) 


.6965 
(direct) 


1.0651 

(M 4 _ 6 -M 4 _ 5 ) 


.5162 

(M 4 _ 7 -M 4 _ 6 ) 


.7298 
(M 4 _ 8 -M 4 _ 7 ) 




1.2413 
(M 3 _ 6 — ikf 4 _ 6 ) 


.9478 
(M 4 _ 6 -M 5 _ 6 ) 


.8136 
(direct) 


.7036 
(M 5 _ 7 -M 5 _ 6 ) 


.6932 
(M 5 _ 8 -M 5 _ 7 ) 




1.3292 
(M 3 _ 7 -M^ 7 ) 


.7606 
(M 4 _ 7 -M 5 _ 7 ) 


.9786 
(M 5 _ 7 -M 6 _ 7 ) 


.5386 
(direct) 


.9426 
(M^-M^) 




? 

(M 3 _ 8 — M 4 _ 8 ) 


.7972 
(M 4 _ 8 -M 5 _ 8 ) 


.7292 

(M 5 _ 8 —M 6 _ 8 ) 


.7248 

(M 6 _ 8 — M 7 _ 8 ) 


.7564 
(direct) 




1.1949 
(direct) 


.5967 
(M 5 _ 3 -M 4 _ 3 ) 


.7127 
(M 6 _ 3 — M 5 _ 3 ) 


.0995 
(M 7 _ 3 -M_ 3 ) 


? 
(M 8 _ 3 — M 7 _ 3 ) 




1.0575 


.7341 
(direct) 


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(M 6 _ 4 -ilf 5 _ 4 ) 


.3613 


.4359 
(M 8 _ 4 -M 7 _ 4 ) 




.8296 


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(M 6 _ 4 -M 6 _ 5 ) 


1.0451 

(direct) 


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(M 7 _ 5 -M 6 _ 5 ) 


.5101 
(M 8 _ 5 -ilf 7 _ 5 ) 




.5678 
(M 7 _ 3 ~M 7 _ 4 ) 


.5264 
(M 7 _ 4 -M 7 _ 5 ) 


.8510 
(M 7 _ h -M 7 _s) 


.5386 
(direct.) 


.7838 
(M 8 _ 6 -M 7 _ 6 ) 




(M 8 _ 3 -M 8 _ 4 ) 


.4522 

(M 8 _ 4 -M 8 _ 5 ) 


.5773 

(M 8 _ 5 — M 8 _ 6 ) 


.8074 

(M 8 _ 6 -M 8 _ 7 ) 


.6350 

(direct) 


Average. . 


1.1479 


.7340 


.8443 


.5359 


.6858 


Weighted 
Average 


1.2008 


.7483 


.8685 


.5606 


.7065 



from M 3 to M 8 is contracted by 0.7235 P.E., an amount which 
is more than some grade intervals (M 6 _ 7 by normal dis., M^ 
and M 7 _ 8 by modified dis.). This is an important difference. 
In the matter of scaling the words, there is, as might be 
supposed, very little difference for the 3d grade — so little as 
to be quite negligible. For the 4th grade there is some difference, 
and for each successive higher grade the difference between 



Derived Forms of Distribution 



103 



the placings of the same word by the two methods becomes 
greater and greater as the asymmetry of the modified curves 
becomes more and more pronounced. 



TABLE XLIV 

Comparison of Averages of Median Distances by Normal Distribu- 
tion and by Modified Distributions 



Normal Distribution 



Modified Distributions 





Unweighted 
Averages 


Weighted 
Averages 


Unweighted 
Averages 


Weighted 
Averages 


M , , 


1 . 3326 
0.8471 
1.0406 
0.6344 
0.9201 


1 . 3505 
0.8363 
1.0505 
0.6608 
0.9101 


1.1479 
0.7340 
0.8443 
0.5359 
0.6858 


1 . 2008 


M A c 


0.7483 


Mil 


0.8685 


M » , 


0.5606 


M„ 


0.7065 


A "■ 7—8 





Table XLV compares the deviations from grade medians of 
the words of the Preferred List by Normal Distribution and 
by Modified Distributions. Figs. 43 to 47 give the same facts 
in graphic form. Words spelled by 50 per cent of pupils are 
of course always at o. Words spelled by more than 50 per cent 
of pupils do not deviate from the median as much by modified 
as by normal distribution. The same is true of those spelled by 
less than 50 per cent of pupils. The easier a word is and the 
harder a word is, the greater, accordingly, is the difference in 
placing. The effect therefore of the modified distributions is to 
shorten the range of the grade scales. In using the scales, 
especially for pupils of the higher grades, all differences in 
ability between individuals or groups would tend to be decreased. 
It seems likely that these differences are in reality more nearly 
what the modified distributions show them to be. The wide 
range of the normal curve especially when its spread is assumed 
to be the same for all grades would seem to extend too far, 
particularly towards the low end. On the other hand, it should 
be said that for the words used in our scale the normal distribu- 
tion gives results that are, practically speaking, satisfactory for 
grade scales. 



io4 Spelling Ability — Its Measurement and Distribution 



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io6 Spelling Ability — Its Measurement mid Distribution 






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io8 Spelling Ability — Its Measurement and Distribution 



Table XLVI and Fig. 48 show a comparison for all grades 
combined. The same shortening of the range is evident but, 
whereas the contraction in the grade scales was more pro- 
nounced at the low ends, it is now in the general scale more 

TABLE XLVI 
The Average Position of Each Word According to Normal Distribu- 
tion and According to Modified Distribution. Point of 
Reference is 3d Grade Median. See Fig. 48 



Word 
No. 



1 

2 
3 
4 
5 

6 

7 

8 

9 

10 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
25 



Word 



even. . . . 
lesson. . . 

only 

smoke . . 
front . . . 

sure 

pear. . . . 
bought . . 
another . 
forty . . . 

pretty . . 
wear. . . . 
button. . 
minute. . 
cousin . . 

nails. . . . 
janitor. . 
saucer . . 
stopping 
sword. . . 

freeze. . . 
touch. . . 
whistle. . 
carriage, 
nor 



Average Position 


Word 


Nor- 


Modi- 


mal 


fied 


No. 


Distri- 


Distri- 




bution 


bution 




.699 


.753 


26 


1.135 


1.018 


27 


.569 


.604 


28 


.835 


.831 


29 


1.057 


.949 


30 


1.568 


1.349 


31 


1.958 


1.697 


32 


1.169 


1.057 


33 


1.078 


1.287 


34 


1.758 


1 . 477 


35 


1.311 


1.137 


36 


1.844 


1.587 


37 


2.026 


1.724 


38 


1.943 


1.687 


39 


1.681 


1.491 


40 


1.379 


1.226 


41 


2.047 


1.773 


42 


2.604 


2.256 


43 


2.213 


1.894 


44 


2.185 


1.766 


45 


1.740 


1.517 


46 


1.709 


1.465 


47 


2.193 


1.870 


48 


2.340 


2.022 


49 


1.652 


1.397 


50 



Word 



already 

beginning. . . 
chicken 

choose 

circus 

grease 

pigeons 

quarrel 

saucy 

tailor 

telegram .... 
telephone. . . 
tobacco 

too 

towel 

Tuesday. . . . 

tying 

whole 

against 

answer 

butcher 

guess 

instead 

raise 

beautiful . . . 



Average Position 



Nor- 
mal 
Distri- 
bution 



699 
917 
897 
502 
141 

294 
739 
069 
666 
866 



2.549 
2.413 

1.988 
3.491 
1.978 

1.550 
1.870 
2.018 
2.106 
1.594 



473 
363 
756 
652 

682 



Modi- 



Distri- 
bution 



2.305 
2.525 

.872 
2.143 
1.872 

2.838 
2.378 
1.816 
2.294 
1.579 

2.204 
2.101 
1.767 
2.998 
1.718 



313 
578 
751 
847 
425 

308 

038 

517 

1.456 

1.519 



evident at the high end. There are also differences in arrange- 
ment, as there could not be in the grade scales. If two words 
which take the same position on the normal scale by ratings 
markedly different in upper and lower grades, but balancing 
each other in the aggregate, these words would not take the 
same position on the modified scale. The one which had 



Derived Forms of Distribution 



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no Spelling Ability — Its Measurement and Distribution 

relatively low ratings in the lower grades would take a position 
above the other whose ratings were relatively high in the lower 
grades. This is because the modified distributions in the upper 
grades are such that counting in from the high end more rapidly 
approaches the median than does counting in the same per cent 
of the area of the normal curve. Take for example the words 
25 nor and 45 answer. Compare the per cent ratings in Table 
XLV. Nor is easy in low grades and hard in upper grades, 
relative to answer. With the same normal distribution for all 
grades nor is a little harder than answer. By the modified dis- 
tributions it is easier. Other words may easily be selected in 
Fig. 48 which show differences in arrangement. It is therefore 
true that the use of modified forms of distribution makes a 
difference which is worth noting in the scaling of words. A 
question to be decided on the evidence of more complete testing 
and a wide use of these forms of distribution is whether the 
differences here shown to exist impair the usefulness of the 
scales we have previously derived. Our judgment at present 
is that they do not. 

§ 20. Conclusions 

We have now certain data in hand and we may make a few 
general statements from them. 

We have selected from a school list of about 5000 words a 
list for test purposes in grades 3-8 which, when put in sentences, 
yielded a list of 270 words. As a result of testing in two schools 
a selected list of 100 words was chosen and to it were added, 
at a later time, 18 more. These were dictated at three schools 
and the 100 words alone subsequently at two more schools. 
From the 118 were chosen two lists of 25 each. The three 
successive selections were made with the purpose of securing 
words which were easy enough in the 3d grade and hard enough 
in the 8th grade to afford a test in those and therefore in inter- 
mediate grades, and which showed regular increases in per cent 
correct from grade to grade. The two 25-word lists were then 
subjected to analysis and found to have high correlations 
between grades and between schools. 

Using the entire test material and the ratings of individual 
pupils and assuming normal distribution and equal variability, 



Conclusion 1 1 1 

the differences between typical grade abilities were found and 
expressed as median intervals. 

The 50 words which had been derived by a threefold selective 
process and subjected to close inspection for permanency as 
between grades and schools were scaled for each grade and for 
all grades combined. By using an "Easy 50- Word List" an 
expression was derived for the zero-point; and, by further test- 
ing under rigidly controlled conditions, previous grade-intervals 
were verified. 

To fill in and extend the scale, the Rice Sentence Test was 
dictated and the word-scores for the Easy 50- Word List were 
used. It is to be understood, however, that neither of these lists 
was subjected to the scrutiny that was made of the Preferred 
List. Accordingly we cannot regard the placing of these words 
as very reliable. 

Finally we have derived and applied tables of frequency 
more or less asymmetrical in character according to the amount 
of retention for each grade and its estimated distribution. By 
using them, results have been obtained which in some instances 
differ considerably from those obtained on the basis of a normal 
distribution. Such differences as appear are, we are convinced, 
differences in the direction of a truer representation of the facts. 
On the whole, however, the differences are not sufficient to 
impair our previous results for any practical use which is likely 
to be made of them. 

It has become evident to us that there is a lack of knowledge 
of the spelling problem not only among teachers but also among 
those who direct their work. This is unfortunate, considering 
the relative definiteness of the subject and the comparative ease 
with which results in it may be scored. Nor is there any special 
consciousness of the need of more insight in this matter. Almost, 
if not quite, all the studies that have hitherto been made have 
dealt with individual performances. The behavior of words has 
received no attention. 

It is our belief, however, that a powerful improvement in the 
teaching of spelling may be derived from a more critical knowl- 
edge and more accurate judgment on the part of teachers and 
supervisors of the material of the subject — i.e., of the words of 
the language. If in a list of 50 words the one word that is 
incontestably hardest is by more than one-fourth of a representa- 



ii2 Spelling Ability — Its Measurement and Distribution 

tive group of teachers judged to be the easiest, or the easiest 
but one, that fact in itself is a very good reason why the word 
is so hard. Pupils misspell it because their teachers do not 
realize the need of teaching it. If text-book makers disagree 
so widely as to put the same words in grades that are three, 
four, and even five years apart, it is proof of the confusion that 
exists as to how hard words are, and when they should be 
taught. There are various types of words, and each type 
requires different treatment. There is the type that does not 
need to be taught at all. There is the type which appears easy in 
the lower classes and (grade considered) hard in the upper 
classes. Such may have been prematurely taught in the lower 
classes. There is the type that appears to possess special difficulty 
for the middle grades. This is due to a constant cause — e.g., in 
the case of whose, to the learning of the use of the apostrophe 
in possessives. There are types of errors ; there is the problem 
of substitution, of illegibility, and of omission. 

To obtain any accurate notion of " word behavior " we must 
rate for words as distinct from individuals, Moreover we must 
give our per cent ratings thus obtained an interpretation for 
difficulty which takes account of the distribution of spelling 
ability. When we do so we shall find how unreliable percentages 
are as indicating differences in difficulty. We shall find, for 
instance, that a difference of 10 per cent between two words 
rated 89 and 99 means more than four times as great a difference 
in difficulty as is that between two words rated at 45 and 55, 
although the percentage difference is in both cases the same. 
Table XLVII (See appendix) is a ready reckoner for the con- 
version of percentages into units that take account of the form 
of distribution, assuming it to be ' Normal.' 

If this study does no more than show the need of word 
criticism and indicate a method, it may be worth while. Every 
school affords a place and every day a time at which something 
may be done to help throw light on the nature of the material 
we deal with in spelling. All such work should be collected and 
made generally available. If teachers, principals, or superin- 
tendents who have made or who hereafter make a study of the 
difficulty of words, will submit them to the author of this study, 
the data will be gratefully received and utilized to disseminate 
a larger and more accurate knowledge. 



APPENDIX 

I. List of Authors and Titles Specifically Referred to 

in the Text 

Thorndike, E. L. ('io). Handwriting. Teachers College Record, Vol. 

XI, No. 2. 
Hillegas, Milo B. ('12). A Scale for the Measurement of Quality in 

English Composition by Young People. Teachers College 

Record, Vol. XIII, No. 4. 
Rice, J. M. ('97). The Futility of the Spelling Grind. Forum, Vol. 

XXIII, pp. 163-172; 409-419. 
Thorndike, E. L. ('13). An Introduction to the Theory of Mental and 

Social Measurements. Second Edition. Teachers College, New 

York. 
Cornman, O. P. (02). Spelling in the Elementary School. Ginn and 

Co., New York. 
Wallin, J. E. Wallace, ('ii). Spelling Efficiency in Relation to Age, 

Grade, and Sex, and the Question of Transfer. Warwick and 

York, Baltimore. 
Pearson, Henry C. ('12). Experimental Studies in the Teaching of 

Spelling. Teachers College Record, Vol. XIII, No. 1. 
Spearman, C. ('06). 'Foot-rule' for Measuring Correlation. Brit. 

Journ. of Psych., Vol. II, Pt. I, July, 1906. 
Brown, William, ('ii). The Essentials of Mental Measurement. Put- 
nam, New York. 
Whipple, Guy Montrose, ('10). Manual of Mental and Physical Tests. 

Warwick and York, Baltimore. 
Klein, Linus W. ('12). A Study in the Psychology of Spelling. Journ. 

of Ed. Psych., Vol. Ill, No. 7. 
Thorndike, E. L. ('07). The Elimination of Pupils from School. 

Bureau of Education, Bulletin No. 4, 1907. 
Ayres, Leonard P. ('09). Laggards in our Schools. Russell Sage 

Foundation, New York. 
Thorndike, E. L. ('10). Promotion, Retardation, and Elimination. 

Psych. Clinic, Vol. Ill, No. 8 and 9. 
Strayer, George Drayton ('ii). Age and Grade Census of Schools and 

Colleges. Bureau of Education, Bulletin No. 5, 191 1. 
Yule, G. Udney ('ii). An Introduction to the Theory of Statistics. 

Lippincott, Philadelphia. 

113 



ii4 Spelling Ability — Its Measurement and Distribution 

II. Lists Referred to in the Text and Used in the Scales 
Preferred List Easy 50-Word 





First 




List 




Rice Sentence List 


I, 


even 


1. 


you 


1. 


running 


44. 


deceive 


2. 


lesson 


2. 


will 


2. 


slipped 


45. 


driving 


3- 


only- 


3- 


hear 


3. 


listened 


46. 


surface 


4- 


smoke 


4- 


him 


4. 


queer 


47- 


rough 


5- 


front 


5. 


coming 


5- 


speech 


48. 


smooth 


6. 


sure 


6. 


he 


6. 


believe 


49- 


hopping 


7- 


pear 


7- 


is 


7- 


weather 


5o. 


certainly 


8. 


bought 


8. 


on 


8. 


changeable 


51. 


grateful 


9- 


another 


C;. 


the 


9- 


whistling 


52. 


elegant 


10. 


forty 


10. 


road 


10. 


frightened 


53- 


present 


11. 


pretty 


II. 


and 


11. 


always 


54- 


patience 


12. 


wear 


12. 


almost 


12. 


changing 


55- 


succeed 


13. 


button 


13. 


sure 


13. 


chain 


56. 


severe 


14. 


minute 


14. 


to 


14. 


loose 


K7. 


accident 


IS- 


cousin 


15. 


pass 


15. 


baking 


58. 


sometimes 


16. 


nails 


It. 


in 


16. 


piece 


59. 


sensible 


17. 


janitor 


17. 


front 


17. 


receive 


60. 


business 


18. 


saucer 


18. 


of 


18. 


laughter 


61. 


answer 


19. 


stopping 


19. 


me 


19. 


distance 


62. 


sweeping 


20. 


sword 


20. 


I 


20. 


choose 


63. 


properly 


21. 


freeze 


21. 


send 


21. 


strange 


64. 


improvement 


22. 


touch 


22. 


for 


22. 


picture 


65. 


fatiguing 


23. 


whistle 


23- 


every 


23. 


because 


66. 


anxious 


24. 


carriage 


24. 


day 


. 24. 


thought 


67. 


appreciate 


25 


nor 


25- 


g° 


25. 


purpose 


68. 


assure 






26. 


into 


26. 


learn 


69. 


imagine 




Second 


27. 


school 


27. 


lose 


70. 


peculiar 


26. 


already 


28. 


but 


28. 


almanac 


7i. 


character 


27. 


beginning 


29. 


do 


29. 


neighbor 


72. 


guarantee 


28. 


chicken 


30. 


not 


30. 


writing 


73- 


approval 


29. 


choose 


31. 


touch 


31. 


language 


74. 


intelligent 


30. 


circus 


32. 


table 


32. 


careful 


73- 


experience 


3i. 


grease 


33- 


also 


33- 


enough 


76. 


delicious 


32. 


pigeons 


34- 


has 


34- 


necessary 


77- 


realize 


33- 


quarrel 


35- 


only 


35- 


waiting 


78. 


importance 


34- 


saucy 


36. 


one 


36. 


disappoint 


79- 


occasion 


35- 


tailor 


37- 


pair 


37. 


often 


80. 


exceptions 


36. 


telegram 


38. 


shoes 


38. 


covered 


81. 


thoroughly 


37- 


telephone 


39- 


they 


39. 


mixture 


82. 


conscientious 


38. 


tobacco 


40. 


are 


40. 


getting 


83. 


therefore 


39- 


too 


41. 


at 


41. 


better 


84. 


ascending 


40. 


towel 


42. 


all 


42. 


feather 


85. 


praise 


41. 


Tuesday 


43- 


pretty 


43- 


light 


86. 


wholesome 


42. 


tying 


44. 


no 










43- 


whole 


45- 


man 










44- 


against 


46. 


ought 










45- 


answer 


47- 


steal 










46. 


butcher 


48. 


even 










47- 


guess 


49. 


a 










48. 


instead 


5o. 


penny 










49. 


raise 














50. 


beautiful 















— 1 



Appendix 115 

III. Memorandum on the Method of Computing with 
Modified Frequency Tables. (Tables XXXVI-XLI.) 

1. Derivation of Median Intervals. Table XLII, lines 4 and 5, gives 
for the 4th grade the number and per cent of pupils who equal or exceed 
the median pupil of each of the other grades. In line 6 the correspond- 
ing P.E. values are shown. These are obtained by using Table XXXVII 
as follows: (a) Since 80.9% of 4th-grade pupils surpass the median 
3d-grade pupil, deduct 8090 cases from the high end of the 4th-grade 
distribution. Since there are 5236.32 above M 4 ( n or. dis.), 2853.68 more 
must be taken, extending to a point which is 1.1079 P.E. below M*(nor. dis.). 
But M 4 (nor. dis.) is itself .087 P.E. below M 4 (mod. dis.). Correcting for this, 
we have 1.1949 P.E. below M^mod. dis.). ( — 1.1079 — .087 = — 1.1949.) 
This is the first entry in line 6 of Table XLII. (b) Deduct 3130 from 
5236.32, leaving 2106.32. By interpellation this corresponds to +.7835 P.E. 
Subtracting .087 P.E. as before, we have +.6965, the second entry in line 
6 of Table XLII. (c) 5236.32 less 11 10 gives 4126.32, corresponding to 
-f 1.8486 P.E. Again subtracting .087 P.E., we have + 1.7616 P.E., which 
is the third entry in line 6, Table XLII. 

2. Scaling _ the Words. For " even," Table XLV, columns headed 
" Modified Distributions," the figures are derived as follows, using for 
each grade its proper frequency table: Third Grade. 59% correct. 
Count out the 5000 highest cases. There are 5 141. 7 above M 3 (nor. dis.). 
We must, therefore, take 758.3 cases below that point. This brings us 
to — .276 P.E. Subtracting (algebraically) .051 P.E., in order to refer 
this to Ms(mod. dis.), we have — .327 P.E., as in Table XLV. Fourth 
Grade. 79% correct. Counting out 7900 cases from the high end, we take 
all the "highs" and 2663.32 of the "lows," reaching as far as — 1.0212 
P.E. But M 4 (mod. dis.) is .087 P.E. above M 4 (nor. dis.). Subtracting this 
amount, we have_ — 1.108 P.E. as in Table XLV. Fifth Grade. When 
percentages are high, it is generally easier to count out their complements 
from the low end. " Even " is in this grade 89% correct. We may there- 
fore count 8900 cases from the high end or 1100 from the low end. In 
either case we reach the 3245th case of the "lows," which corresponds 
to — 1,306 P.E. Correcting for the deflection of the median from its 
"normal" position (.215 P.E.), we have — 1,521 P.E. as given. Sixth 
Grade. 3696 — 700 = 2996. The 2996th case corresponds to — 1.339 P-E- 
Median displacement =.418 P.E. Subtracting from — 1-339 P.E., we 
have — 1-757 P-E., as given. The 7th and 8th grade positions are derived 
in the same way, care being taken to use the proper grade table of fre- 
quency in each case. 

Table XLVI. The average position for each word as given in the 
column headed " Modified Distributions " was computed as follows : Add 
to the P.E. value of " even " for each grade (Table XLV) the distance 
which the grade median is above the 3rd-grade median. From Table 
XLII these distances are shown to be: M 3 -«, 1.148 P. E. ; M 3 - 5 , 1.882 P.E. ; 
M3-6, 2.726 P.E. ; Afs-r, 3.262 P.E.; Mas. 3.948 P.E. Adding these values 
to those of Table XLV, beginning with the 4th grade and writing the 
3rd grade as given, we have the following P.E. values: —.327, +040, 
+.361, +.969, -f 1.541, and + 1.932. The average of these is +.753 P-E., 
as given for the word " even " in Table XLVI. The average positions of 
the remaining words were computed in the same way. 



n6 Appendix 

IV. TABLE XL VII — P.E. Values Corresponding to Given Per Cents 
of the Normal Surface of Frequency, Per Cents Being Taken 
from the Median 








.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 





.000 


.004 


.007 


.011 


.015 


.019 


.022 


.026 


030 


.033 


1 


.037 


.041 


.044 


.048 


.052 


.056 


.059 


.063 


067 


.071 


2 


.074 


.078 


.082 


.085 


.089 


.093 


.097 


.100 


104 


.108 


3 


.112 


.115 


.119 


.123 


.127 


.130 


.134 


.138 


.141 


.145 


4 


.149 


.153 


.156 


.160 


.164 


.168 


.172 


.175 


179 


.183 


5 


.187 


.190 


.194 


.198 


.201 


.205 


.209 


.213 


216 


.220 


6 


.224 


.228 


.231 


.235 


.239 


.243 


.246 


.250 


.254 


.258 


7 


.261 


.265 


.269 


.273 


.277 


.280 


.284 


.288 


.292 


.296 


8 


.299 


.303 


.307 


.311 


.315 


.318 


.322 


.326 


.330 


.334 


9 


.337 


.341 


.345 


.349 


.353 


.357 


.360 


.364 


.368 


.372 


10 


.376 


.380 


.383 


.387 


.391 


.395 


.399 


.403 


.407 


.410 


11 


.414 


.418 


.422 


.426 


.430 


.434 


.437 


.441 


.445 


.449 


12 


.453 


.457 


.461 


.464 


.468 


.472 


.476 


.480 


.484 


.489 


13 


.492 


.496 


.500 


.504 


.508 


.512 


.516 


.519 


.523 


.527 


14 


.531 


.535 


.539 


.543 


.547. 


551 


.555 


.559 


.563 


.567 


15 


.571 


.575 


.579 


.583 


.588 


.592 


.596 


.600 


.603 


.608 


16 


.612 


.616 


.620 


.624 


.628 


.632 


.636 


.640 


.644 


.648 


17 


.652 


.656 


.660 


.665 


.669 


.673 


.677 


.681 


.685 


.689 


18 


.693 


.698 


.702 


.706 


.710 


.714 


.719 


.723 


.727 


.731 


19 


.735 


.740 


.744 


.748 


.752 


.756 


.761 


.765 


.769 


.773 


20 


.778 


.782 


.786 


.790 


.795 


.799 


.803 


.807 


.812 


.816 


21 


.820 


.825 


.829 


.834 


.838 


.842 


.847 


.851 


.855 


.860 


22 


.864 


.869 


.873 


.878 


.882 


.886 


.891 


.895 


.900 


.904 


23 


.909 


.913 


.918 


.922 


.927 


.931 


.936 


.940 


.945 


.949 


24 


.954 


.958 


.963 


.968 


.972 


.977 


.982 


.986 


.991 


.996 


25 


1.000 


1.005 


1.009 


1.014 1.019 


1.024 


1.028 


1.033 1 


.038 


1.042 


26 


1.047 


1.052 


1.057 


1.062 


1.067 


1.071 


1.076 


1.081 1 


.086 


1.091 


27 


1.096 


1.101 


1.105 


1.110 


1.115 


1.120 


1.125 


1.130 1 


.135 


1.140 


28 


1.145 


1.150 


1.155 


1.160 


1.165 


1.170 


1.176 


1.181 1 


.186 


1.191 


29 


1.196 


1.201 


1.206 


1.211 


1.217 


1.222 


1.227 


1.232 1 


.238 


1.243 


30 


1.248 


1.253 


1.259 


1.264 


1.269 


1.275 


1.279 


1.286 1 


.291 


1.296 


31 


1.302 


1.307 


1.313 


1.318 


1.324 


1.329 


1.335 


1.340 1 


.346 


1.351 


32 


1.357 


1.363 


1.368 


1.374 


1.380 


1.386 


1.391 


1.397 1 


.403 


1.409 


33 


1.415 


1.421 


1.427 


1.432 


1.438 


1.444 


1.450 


1.456 1 


.462 


1.469 


34 


1.475 


1.481 


1.487 


1.493 


1.499 


1.506 


1.512 


1.518 1 


.524 


1.531 


35 


1.537 


1.543 


1.549 


1.556 


1.563 


1.569 


1.576 


1.582 1 


.589 


1.595 


36 


1.602 


1.609 


1.616 


1.622 


1.629 


1.636 


1.643 


1.649 1 


.656 


1.663 


37 


1.670 


1.677 


1.685 


1.692 


1.699 


1.706 


1.713 


1.720 1 


.728 


1.735 


38 


1.742 


1.749 


1.757 


1.765 


1.772 


1.780 


1.788 


1.795 1 


.803 


1.811 


39 


1.819 


1.827 


1.835 


1.843 


1.851 


1.859 


1.867 


1.875 1 


.884 


1.892 


40 


1.900 


1.909 


1.918 


1.926 


1.935 


1.944 


1.953 


1.962 1 


.971 


1.979 


41 


1.988 1.997 2.007 2.016 2.026 2.035 2.044 2.054 2.064 2.074 


42 


2.083 


2.093 


2.103 2.114 2.124 


2.134 2.145 


2.155 2 


.166 


2.177 


43 


2.188 2.199 2.211 


2.222 


2.234 


2.245 2.257 2.269 2 


.281 


2.293 


44 


2.305 2.318 


2.331 


2.344 


2.357 2.370 


2.384 2.397 2 


.411 


2.425 


45 


2.439 


2.453 


2.468 


2.483 


2.498 


2.514 


2.530 


2.546 2 


.562 


2.579 


46 


2.597 2.614 


2.631 


2.648 


2.667 2.686 


2.706 


2.726 2 


.746 2.767 


47 


2.789 


2.811 


2.834 2.857 2.881 


2.905 


2.932 


2.958 2 


.986 


3.015 


48 


3.044 3.077 3.111 


3.146 


3.182 


3.219 


3.258 


3.300 3 


.346 3.395 


49 
50 


3.450 3.506 3.571 3.643 3.725 3.820 3.938 4.083 4.275 4.600 



VITA 

Burdette Ross Buckingham, born at Riverhead, N. Y., 
Dec. g } 1876. 

A.B., Wesleyan University, Middletown, Conn., 1899. 
Graduate student, Wesleyan University, 1899- 1900. M.A. 
(Major: English), 1900. Student, Albany State Normal Col- 
lege, 1900-1901. Pd.B., 1901. 

Principal, Northport High School, Northport, N. Y., 
1901-1904. Assistant teacher, mathematics, Morris High School, 
New York City, 1904- 1907. Principal, Public School 86, 
Borough of Queens, 1907-. 

Graduate student, Teachers College, Columbia University, 
1905-1913. 

117 



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